# Constructing Disjoint Steiner Trees in Sierpiński Graphs††thanks: Supported by the National Natural Science Foundation of China (Nos. 12201375, 12471329 and 12061059) and the Qinghai Key Laboratory of Internet of Things Project (2017-ZJ-Y21).
**Authors**:
- Chenxu Yang (School of Computer)
- Qinghai Normal University
- Xining
- Qinghai
- Ping Li (School of Mathematics and Statistics)
- Shaanxi Normal University
- Xi’an
- Shaanxi
- Yaping Mao
- Xining
- Qinghai
- Eddie Cheng (Department of Mathematics)
- Oakland University
- Rochester
- MI USA
- Ralf Klasing
- Université de Bordeaux
- Bordeaux INP
- CNRS
- LaBRI
- UMR
- Talence
> Corresponding author.
25 2 10.46298/fi.12478
## Abstract
Let $G$ be a graph and $S\subseteq V(G)$ with $|S|\geq 2$ . Then the trees $T_{1},T_{2},\cdots,T_{\ell}$ in $G$ connecting $S$ are internally disjoint Steiner trees (or $S$ -Steiner trees) if $E(T_{i})\cap E(T_{j})=\emptyset$ and $V(T_{i})\cap V(T_{j})=S$ for every pair of distinct integers $1\leq i,j\leq\ell$ . Similarly, if we only have the condition $E(T_{i})\cap E(T_{j})=\emptyset$ but without the condition $V(T_{i})\cap V(T_{j})=S$ , then they are edge-disjoint Steiner trees $S$ -Steiner trees. The generalized $k$ -connectivity, denoted by $\kappa_{k}(G)$ , of a graph $G$ , is defined as $\kappa_{k}(G)=\min\{\kappa_{G}(S)|S\subseteq V(G)\ \textrm{and}\ |S|=k\}$ , where $\kappa_{G}(S)$ is the maximum number of internally disjoint $S$ -Steiner trees. The generalized $k$ -edge-connectivity $\lambda_{k}(G)$ of $G$ is defined as $\lambda_{k}(G)=\min\{\lambda_{G}(S)\,|\,S\subseteq V(G)\ and\ |S|=k\}$ , where $\lambda_{G}(S)$ is the maximum number of edge-disjoint Steiner trees connecting $S$ in $G$ . These concepts are generalizations of the concepts of connectivity and edge-connectivity, and they can be used as measures of vulnerability of networks. It is, in general, difficult to compute these generalized connectivities. However, there are precise results for some special classes of graphs. In this paper, we obtain the exact value of $\lambda_{k}(S(n,\ell))$ for $3\leq k\leq\ell^{n}$ , and the exact value of $\kappa_{k}(S(n,\ell))$ for $3\leq k\leq\ell$ , where $S(n,\ell)$ is the Sierpiński graphs with order $\ell^{n}$ . As a direct consequence, these graphs provide additional interesting examples when $\lambda_{k}(S(n,\ell))=\kappa_{k}(S(n,\ell))$ . We also study the some network properties of Sierpiński graphs. Keywords: Steiner Tree; Generalized Connectivity; Sierpiński Graph. AMS subject classification 2010: 05C40, 05C85.
volume: 194 issue: 1
Constructing disjoint Steiner trees in Sierpiński graphs
## 1 Introduction
All graphs considered in this paper are undirected, finite and simple. We refer the readers to [1] for graph theoretical notation and terminology not described here. For a graph $G$ , let $V(G)$ and $E(G)$ denote the set of vertices and the set of edges of $G$ , respectively. The neighborhood set of a vertex $v\in V(G)$ is $N_{G}(v)=\{u\in V(G)\,|\,uv\in E(G)\}$ . The degree of a vertex $v$ in $G$ is denoted by $d(v)=|N_{G}(v)|$ . Denote by $\delta(G)$ ( $\Delta(G)$ ) the minimum degree (maximum degree) of the graph $G$ . For a vertex subset $S\subseteq V(G)$ , the subgraph induced by $S$ in $G$ is denoted by $G[S]$ and similarly $G[V\setminus S]$ for $G\setminus S$ or $G-S$ . Especially, $G-v$ is $G[V\setminus\{v\}]$ . Let $\overline{G}$ be the complement of $G$ . For a partition $\mathcal{P}=\{V_{1},V_{2},\ldots,V_{t}\}$ of $V(G)$ , let $G/\mathcal{P}$ be the graph obtained from $G$ by deleting $\bigcup_{i\in[t]}E(G[V_{i}])$ and then identifying each $V_{i}$ , respectively. For any positive integers $n$ , we always use the convenient notation $[n]$ to denote the set $\{1,2,\cdots,n\}$ .
### 1.1 Generalized (edge-)connectivity
Connectivity and edge-connectivity are two of the most basic concepts of graph-theoretic measures. Such concepts can be generalized, see, for example, [16]. For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of at least two vertices, an $S$ -Steiner tree or a Steiner tree connecting $S$ (or simply, an $S$ -tree) is a subgraph $T=(V^{\prime},E^{\prime})$ of $G$ that is a tree with $S\subseteq V^{\prime}$ . Note that when $|S|=2$ a minimal $S$ -Steiner tree is just a path connecting the two vertices of $S$ .
Let $G$ be a graph and $S\subseteq V(G)$ with $|S|\geq 2$ . Then the trees $T_{1},T_{2},\cdots,T_{\ell}$ in $G$ are internally disjoint $S$ -trees if $E(T_{i})\cap E(T_{j})=\emptyset$ and $V(T_{i})\cap V(T_{j})=S$ for every pair of distinct integers $i,j$ , $1\leq i,j\leq\ell$ . Similarly, if we only have the condition $E(T_{i})\cap E(T_{j})=\emptyset$ but without the condition $V(T_{i})\cap V(T_{j})=S$ , then they are edge-disjoint $S$ -trees (Note that while we do not have the condition $V(T_{i})\cap V(T_{j})=S$ , it is still true that $S\subseteq V(T_{i})\cap V(T_{j})$ as $T_{i}$ and $T_{j}$ are $S$ -trees.) The generalized $k$ -connectivity, denoted by $\kappa_{k}(G)$ , of a graph $G$ , is defined as $\kappa_{k}(G)=\min\{\kappa_{G}(S)|S\subseteq V(G)\ \textrm{and}\ |S|=k\}$ , where $\kappa_{G}(S)$ is the maximum number of internally disjoint $S$ -trees. The generalized local edge-connectivity $\lambda_{G}(S)$ is the maximum number of edge-disjoint $S$ -trees in $G$ . The generalized $k$ -edge-connectivity $\lambda_{k}(G)$ of $G$ is defined as $\lambda_{k}(G)=\min\{\lambda_{G}(S)\,|\,S\subseteq V(G)\ and\ |S|=k\}$ . Since internally disjoint $S$ -trees are edge-disjoint but not vice versa, it follows from the definitions that $\kappa_{k}(G)\leq\lambda_{k}(G)$ . There are many results on generalized (edge-)connectivity; see the book [15] by Li and Mao.
For a graph $G$ and two distinct vertices $x,y$ of $G$ , the local connectivity $p_{G}(x,y)$ of $x$ and $y$ is defined as the maximum number of pairwise internally disjoint paths between $x$ and $y$ , and the local edge-connectivity $\lambda_{G}(x,y)$ is defined as the maximum number of pairwise edge-disjoint paths between $x$ and $y$ . The connectivity of $G$ is defined as $\kappa(G)=\min\{p_{G}(x,y)\,|\,x,y\in V(G),\,x\neq y\}$ , and the edge-connectivity of $G$ is defined as $\lambda(G)=\min\{\lambda_{G}(x,y)\,|\,x,y\in V(G),\,x\neq y\}$ . It is clear that when $|S|=2$ , $\lambda_{2}(G)$ is just the standard edge-connectivity $\lambda(G)$ of $G$ , $\kappa_{2}(G)=\kappa(G)$ , that is, the standard connectivity of $G$ . Thus $\kappa_{k}(G)$ and $\lambda_{k}(G)$ are the generalized connectivity of $G$ and the generalized edge-connectivity of $G$ , respectively.
As it is well-known, for any graph $G$ , we have polynomial-time algorithms to find the classical connectivity $\kappa(G)$ and edge-connectivity $\lambda(G)$ . Given two fixed positive integers $k$ and $\ell$ , for any graph $G$ the problem of deciding whether $\lambda_{k}(G)\geq\ell$ can be solved by a polynomial-time algorithm. If $k\ (k\geq 3)$ is a fixed integer and $\ell\ (\ell\geq 2)$ is an arbitrary positive integer, the problem of deciding whether $\kappa(S)\geq\ell$ is $NP$ -complete. For any fixed integer $\ell\geq 3$ , given a graph $G$ and a subset $S$ of $V(G)$ , deciding whether there are $\ell$ internally disjoint Steiner trees connecting $S$ , namely deciding whether $\kappa(S)\geq\ell$ , is $NP$ -complete. For more details on the computational complexity of generalized connectivity and generalized edge-connectivity, we refer to the book [15].
In addition to being a natural combinatorial measure, generalized $k$ -connectivity can be motivated by its interesting interpretation in practice. For example, suppose that $G$ represents a network. If one wants to “connect” a pair of vertices of $G$ “minimally”, then a path is used to “connect” them. More generally, if one wants to “connect” a set $S$ of vertices of $G$ , with $|S|\geq 3$ , “minimally”, then it is desirable to use a tree to “connect” them. Such trees are precisely $S$ -trees, which are also used in computer communication networks (see [8]) and optical wireless communication networks (see [6]).
From a theoretical perspective, generalized edge-connectivity is related to Nash-Williams-Tutte theorem and Menger theorem; see [15]. The generalized edge-connectivity has applications in $VLSI$ circuit design. In this application, a Steiner tree is needed to share an electronic signal by a set of terminal nodes. Another application, which is our primary focus, arises in the Internet Domain. Imagine that a given graph $G$ represents a network. We arbitrarily choose $k$ vertices as nodes. Suppose one of the nodes in $G$ is a broadcaster, and all other nodes are either users or routers (also called switches). The broadcaster wants to broadcast as many streams of movies as possible, so that the users have the maximum number of choices. Each stream of movie is broadcasted via a tree connecting all the users and the broadcaster. So, in essence we need to find the maximum number Steiner trees connecting all the users and the broadcaster, namely, we want to get $\lambda(S)$ , where $S$ is the set of the $k$ nodes. Clearly, it is a Steiner tree packing problem. Furthermore, if we want to know whether for any $k$ nodes the network $G$ has above properties, then we need to compute $\lambda_{k}(G)=\min\{\lambda(S)\}$ in order to prescribe the reliability and the security of the network. For more details, we refer to the book [15].
### 1.2 Sierpiński graphs
In 1997, Klavžar and Milutinović introduced the concept of Sierpiński graph $S(n,\ell)$ in [11]. We denote $n$ -tuples $V^{n}$ by the set
$$
V^{n}=\{\langle u_{0}u_{1}\cdots u_{n-1}\rangle|\ u_{i}\in\{0,1,\ldots,\ell-1
\}\textrm{ and }i\in\{0,1,\ldots,n-1\}\}.
$$
A word $u$ of size $n$ are denoted by $\langle u_{0}u_{1},\cdots,u_{n-1}\rangle$ in which $u_{i}\in\{0,\ldots,\ell-1\}$ . The concatenation of two words $u=\langle u_{0}u_{1}\cdots u_{n-1}\rangle$ and $v=\langle v_{0}v_{1}\cdots v_{n-1}\rangle$ is denoted by $uv$ .
**Definition 1**
*The Sierpiński graph $S(n,\ell)$ is defined as below, for $n\geq 1$ and $\ell\geq 3$ , the vertex set of $S(n,\ell)$ consists of all $n$ -tuples of integers $0,1,\cdots,\ell-1$ . That is, $V(S(n,\ell))=V^{n}$ , where $uv=(u_{0}u_{1}\cdots u_{n-1},v_{0}v_{1}\cdots v_{n-1})$ is an edge of $E(S(n,\ell))$ if and only if there exists $d\in\{0,1,\ldots,\ell-1\}$ such that: $(1)$ $u_{j}=v_{j}$ , if $j<d$ ; $(2)$ $u_{d}\neq v_{d}$ ; $(3)$ $u_{j}=v_{d}$ and $v_{j}=u_{d}$ , if $j>d$ .*
<details>
<summary>x1.png Details</summary>
### Visual Description
## Diagram: Hierarchical Network Structure
### Overview
The image depicts a hierarchical network diagram with nodes labeled numerically. The structure consists of three distinct layers: a top tier, a central tier, and a bottom tier. Nodes are interconnected via edges, forming a tree-like hierarchy with additional lateral connections in the bottom tier.
### Components/Axes
- **Nodes**: Labeled with two-digit numbers (e.g., 12, 13, 11, 10, 01, 02, 00, 03, 21, 22, 20, 23, 30, 33, 31, 32).
- **Edges**: Lines connecting nodes, representing relationships or pathways.
- **Hierarchy**:
- **Top Tier**: Nodes 12, 13, 11, 10, 01.
- **Central Tier**: Nodes 02, 00, 03.
- **Bottom Tier**: Nodes 21, 22, 20, 23 (left), and 30, 33, 31, 32 (right).
### Detailed Analysis
1. **Top Tier**:
- Node 12 connects to 13 and 11.
- Node 13 connects to 11 and 12.
- Node 11 connects to 12, 13, and 10.
- Node 10 connects to 11 and 01.
- Node 01 connects to 10, 02, 00, and 03.
2. **Central Tier**:
- Node 02 connects to 01, 21, and 22.
- Node 00 connects to 01, 20, and 30.
- Node 03 connects to 01, 33, and 31.
3. **Bottom Tier**:
- **Left Subnetwork**:
- Node 21 connects to 22 and 20.
- Node 22 connects to 21, 20, and 23.
- Node 20 connects to 21, 22, and 23.
- Node 23 connects to 22 and 20.
- **Right Subnetwork**:
- Node 30 connects to 33 and 31.
- Node 33 connects to 30, 31, and 32.
- Node 31 connects to 30, 33, and 32.
- Node 32 connects to 33 and 31.
### Key Observations
- **Hierarchical Flow**: The top tier (12–01) forms a descending chain, while the central tier (02–00–03) branches out to the bottom tier.
- **Cyclic Connections**: The bottom tier nodes form closed loops (e.g., 21–22–20–23 and 30–33–31–32), suggesting feedback or redundancy.
- **Central Hub**: Node 00 acts as a critical junction, connecting the central tier to both left and right subnetworks.
### Interpretation
This diagram likely represents a system with a centralized control node (00) managing distributed subsystems. The top tier may symbolize high-level decision-making (e.g., strategic planning), while the bottom tier reflects operational units. The cyclic connections in the bottom tier imply interdependent processes or resource sharing. Node 01 serves as a bridge between strategic and operational layers, highlighting its role in information flow. The absence of explicit labels beyond numerical identifiers suggests a generic or abstract model, possibly for organizational structure, data flow, or network topology.
</details>
Figure 1: $S(2,4)$
Sierpiński graph $S(2,4)$ is shown in Figure 1. Note that $S(n,\ell)$ can be constructed recursively as follows: $S(1,\ell)$ is isomorphic to $K_{\ell}$ , which vertex set is 1-tuples set $\{0,\ldots,\ell-1\}$ . To construct $S(n,\ell)$ for $n>1$ , we take copies of $\ell$ times $S(n-1,\ell)$ and add the letter $i$ on the top of the vertices in $i$ -th copy of $S(n-1,\ell)$ , denoted by $S^{i}(n,\ell)$ . Note that there is exactly one edge (bridge edge) between $S^{i}(n,\ell)$ and $S^{j}(n,\ell)$ , $i\neq j$ , namely the edge between vertices $\langle ij\cdots j\rangle$ and $\langle ji\cdots i\rangle$ .
The vertices $(\underbrace{i,i,\cdots,i}_{n})$ , $i\in\{0,1,\ldots,\ell-1\}$ are the extreme vertices of $S(n,\ell)$ . Note that an extreme vertex $u$ of $S(n,\ell)$ has degree $d(u)=\ell-1$ . For $i\in\{0,\ldots,\ell-1\}$ and $n\geq 2$ , let $S^{i}(n-1,\ell)$ denote the subgraph of $S(n,\ell)$ induced by the vertices of the form $\{\langle iu_{1}\cdots u_{n-1}\rangle\,|\,0\leq u_{i}\leq\ell-1\}$ . The vertex set $V(S(n,\ell))$ can be partitioned into $\ell$ parts $V(S^{0}(n-1,\ell)),V(S^{1}(n-1,\ell)),\ldots,V(S^{\ell-1}(n-1,\ell))$ . For each $0\leq i\leq\ell-1$ , $S^{i}(n-1,\ell)$ is isomorphic to $S(n-1,\ell)$ . Note that $V(S(n,\ell))=V(S^{0}(n-1,\ell))\cup\cdots\cup V(S^{\ell-1}(n-1,\ell))$ and $S(n,\ell)$ is the graph obtained from $S^{0}(n-1,\ell),\ldots,S^{\ell-1}(n-1,\ell)$ by adding exactly one edge (bridge edge) between $S^{i}(n-1,\ell)$ and $S^{j}(n-1,\ell)$ , $i\neq j$ , and the bridge edge joins $\langle ij\cdots j\rangle$ and $\langle ji\cdots i\rangle$ (notices that $\langle ij\cdots j\rangle$ and $\langle ji\cdots i\rangle$ are extreme vertices of $S^{i}(n-1,\ell)$ and $S^{j}(n-1,\ell)$ , respectively, if we regard $S^{i}(n-1,\ell)$ and $S^{j}(n-1,\ell)$ as two copies of $S(n-1,\ell)$ ).
Sierpiński graphs generalize Hanoi graphs which can be viewed as “discrete” finite versions of a Sierpiński gasket [23, 10]. Xue considered the Hamiltionicity and path $t$ -coloring of Sierpiński-like graphs in [25]; furthermore, they proved that $Val(S(n,k))=Val(S[n,k])=\lfloor k/2\rfloor$ , where $Val(S(n,k))$ is the linear arboricity of Sierpiński graphs. We remark that although Sierpiński graphs are not regular, they are “almost” regular as the extreme vertices have degrees one less than the degrees of non-extreme vertices.
### 1.3 Preliminaries and our results
Chartrand et al. [4] and Li et al. [16] obtained the exact value of $\kappa_{k}(K_{n})$ .
**Theorem 1.1 ([16,4])**
*For every two integers $n$ and $k$ with $2\leq k\leq n$ ,
$$
\kappa_{k}(K_{n})=\lambda_{k}(K_{n})=n-\lceil k/2\rceil.
$$*
The following result is on the Hamiltonian decomposition of Sierpiński graphs.
**Theorem 1.2**
*[25] $(1)$ For even $\ell\geq 2$ , $S(n,\ell)$ can be decomposed into edge disjoint union of $\frac{\ell}{2}$ Hamiltonian paths of which the end vertices are extreme vertices. $(2)$ For odd $\ell\geq 3$ , there exist $\frac{\ell-1}{2}$ edge-disjoint Hamiltonian paths whose two end vertices are extreme vertices in $S(n,\ell)$ .*
In fact, Theorem 1.2 is used in the proof of Theorem 1.5, which give an lower bound for the generalized $k$ -edge connectivity of Sierpiński graphs $S(n,\ell)$ . We require the following result.
**Theorem 1.3 ([5,26])**
*Suppose that $G$ is a complete graph with $V(G)=\{v_{0},\cdots,v_{N-1}\}$ . If $N=2n$ , then $G$ can be decomposed into $n$ Hamiltonian paths
$$
\{i\in\{1,2,\ldots,n\}:\ L_{i}=v_{0+i}v_{1+i}v_{2n-1+i}v_{2+i}v_{2n-2+i}\cdots
v
_{n+1+i}v_{n+1}\},
$$
where the subscripts take modulo $2n$ . If $N=2n+1$ , then $G$ can be decomposed into $n$ Hamiltonian paths
$$
\{i\in\{1,2,\ldots,n\}:\ L_{i}=\ v_{0+i}v_{1+i}v_{2n+i}v_{2+i}v_{2n-1+i}\cdots
v
_{n+i}v_{n+1+i}\}
$$
and a matching $M=\{v_{n-i}v_{n+1}:\ i\in\{1,2,\ldots,n\}\}$ , where the subscripts take modulo $2n+1$ .*
The following result is derived from Theorem 1.3, and we will use it later.
**Corollary 1.4**
*Let $s$ be an integer with $s\leq\frac{N}{2}$ . Suppose that $G$ is the complete graph with $V(G)=\{v_{1},\cdots,v_{N}\}$ and $\mathcal{S}=\{\ \{v_{i_{1}},v_{i_{2}}\}:i\in\{1,2,\ldots,s\}\}$ is a collection of pairwise disjoint $2$ -subsets of $V(G)$ . Then there are $s$ edge-disjoint Hamiltonian paths $L_{1},\cdots,L_{s}$ such that $v_{i_{1}},v_{i_{2}}$ are endpoints of $L_{i}$ .*
Our main result is as follows.
**Theorem 1.5**
*$(i)$ For $3\leq k\leq\ell$ , we have
$$
\kappa_{k}(S(n,\ell))=\lambda_{k}(S(n,\ell))=\ell-\left\lceil k/2\right\rceil.
$$ $(ii)$ For $\ell+1\leq k\leq{\ell}^{n}$ , we have
$$
\kappa_{k}(S(n,\ell))\leq\left\lfloor\ell/2\right\rfloor\mbox{ and }\lambda_{k
}(S(n,\ell))=\left\lfloor\ell/2\right\rfloor.
$$*
## 2 Proof of Theorem 1.5
This section is organized as follows. We first introduce some notations, operations and auxiliary graphs that are used in the proof. Then we give a transitional theorem (Theorem 2.1), which is the main tool for the proof of Theorem 1.5. Theorem 2.1 will be proved by constructing an algorithm, and the proof takes up almost entire Section 2 (the proof is divided into two parts: the construction of algorithm and the proof of its correctness). Finally, we complete the proof of Theorem 1.5 by using Theorem 2.1.
### 2.1 Notations, operations and auxiliary graphs
For $1\leq s<n$ , by the definition of the Sierpiński graph $S(n,\ell)$ , we have that $G=S(n,\ell)$ consists of $\ell^{n-s}$ Sierpiński graphs $S(s,\ell)$ , and we call each such $S(s,\ell)$ an $s$ -atom of $G$ . Note that $\mathcal{P}=\{V(H):\ H\mbox{ is an }s\mbox{-atom of }G\}$ is a partition of $V(G)$ . We use $G_{s}$ to denote the graph $G/\mathcal{P}$ . It is obvious that $G_{s}$ is a simple graph and $G_{s}=S(n-s,\ell)$ . Note that $G_{n}$ is a single vertex and $G_{0}=G$ . For a vertex $u$ of $G_{s}$ , we use $A_{s,u}$ to denote the $s$ -atom of $G$ which is contracted into $u$ . See Figures 4, 4 and 4 for illustrations.
| Notation | Corresponding interpretation |
| --- | --- |
| $s$ -atom | A copy of $S(s,\ell)$ in $G=S(n,\ell)$ , where $0\leq s\leq n$ . |
| $G_{s}$ | The graph obtained from $G$ by contracting all $\ell^{n-s}$ $s$ -atoms in $G$ . |
| $A_{u,s}$ | The $s$ -atom in $G$ which is contracted into $u$ , where $u\in V(G_{s})$ . |
| $H^{u}$ | $G_{s-1}$ can be obtained form $G_{s}$ by expending each $u\in V(G_{s})$ to a complete graph $H^{u}=K_{\ell}$ . |
| $u^{e},v^{e}$ | If $e=uv$ is an edge of $G_{s}$ , then $e$ is also an edge of $G_{s-1}$ . let $u_{e},v_{e}$ denote endpoints of $e$ in $G_{s-1}$ such that $u_{e}\in H^{u}$ and $v_{e}\in H^{v}$ . |
| $U$ | The set of labelled vertices in $G$ ( $|U|=k$ ). |
| $U^{s}$ | The set of labelled vertices in $G_{s}$ . |
| $W^{u}$ | The set of labelled vertices in $H^{u}$ . |
Table 1: Notations for and their meanings.
Note that $G_{s-1}$ is the graph obtained from $G_{s}$ by replacing each vertex $u$ with a complete graph $K_{\ell}$ . We denote by $H^{u}$ the complete graph replacing $u$ (see Figures 4 and 4 for illustrations). For an edge $e=uv$ of $G_{s}$ , $e$ is also an edge of $G_{s-1}$ . We use $u_{e}$ and $v_{e}$ to denote the ends of $e$ in $G_{s-1}$ , respectively, such that $u_{e}\in V(H^{u})$ and $v_{e}\in V(H^{v})$ . (See Figure 5 for an illustration.)
Fix a subset $U$ of $V(G)$ arbitrary such that $|U|=k$ , where $3\leq k\leq\ell$ . For the graph $G_{s}$ , a vertex $u$ of $G_{s}$ is labelled if $V(A_{s,u})\cap U\neq\emptyset$ , and $u$ is unlabelled otherwise. We use $U^{s}$ to denote the set of labelled vertices of $G_{s}$ . For a labelled vertex $u$ of $G_{s}$ , let $W^{u}=V(H^{u})\cap U^{s-1}$ , that is, the set of labelled vertices in $H^{u}$ . See Figure 4 as an example, if $U=\{u_{000},u_{001},u_{013},u_{122}\}$ , then $U^{1}=\{u_{00},u_{01},u_{12}\}$ and $W^{u_{0}}=\{u_{00},u_{01}\}$ in Figure 4. For ease of reading, above notations and their meanings are summarized in Table 1.
We construct an out-branching $\overrightarrow{T}$ with vertex set $V(\overrightarrow{T})=\bigcup_{i\in\{0,1,\ldots,n\}}\{x:\ x\in V(G_{i})\}$ and arc set $A(\overrightarrow{T})=\{(x,y):\ y\in V(H^{x})\}$ . The root of $\overrightarrow{T}$ is denoted by $v_{root}$ (it is clear that $V(G_{n})=\{v_{root}\}$ ). For two different vertices $x,y$ of $V(\overrightarrow{T})$ , if there is a direct path from $x$ to $y$ , then we say that $x\prec y$ , and denote the directed path by $x\overrightarrow{T}y$ . The following result is straightforward.
**Fact 1**
*If $x_{i}\in V(\overrightarrow{T})$ for $i\in[p]$ and $x_{1}\prec x_{2}\prec\ldots\prec x_{p}$ , then $\sum_{i\in[p]}|W^{x_{i}}|\leq k+p-1$ .*
$u_{000}$ $u_{001}$ $u_{002}$ $u_{003}$ $u_{010}$ $u_{012}$ $u_{011}$ $u_{013}$ $u_{020}$ $u_{021}$ $u_{030}$ $u_{031}$ $u_{022}$ $u_{023}$ $u_{032}$ $u_{033}$ $u_{100}$ $u_{101}$ $u_{110}$ $u_{111}$ $u_{102}$ $u_{103}$ $u_{112}$ $u_{113}$ $u_{120}$ $u_{121}$ $u_{130}$ $u_{131}$ $u_{122}$ $u_{123}$ $u_{132}$ $u_{133}$ $u_{200}$ $u_{201}$ $u_{210}$ $u_{211}$ $u_{300}$ $u_{301}$ $u_{310}$ $u_{311}$ $u_{202}$ $u_{203}$ $u_{212}$ $u_{213}$ $u_{302}$ $u_{303}$ $u_{312}$ $u_{313}$ $u_{220}$ $u_{221}$ $u_{230}$ $u_{231}$ $u_{320}$ $u_{321}$ $u_{330}$ $u_{331}$ $u_{222}$ $u_{223}$ $u_{232}$ $u_{233}$ $u_{322}$ $u_{323}$ $u_{332}$ $u_{333}$ $A(2,u_{0})$ $A(2,u_{1})$
Figure 2: $G=G_{0}=S(3,4)$
$u_{00}$ $u_{01}$ $u_{02}$ $u_{03}$ $u_{10}$ $u_{11}$ $u_{12}$ $u_{13}$ $u_{20}$ $u_{21}$ $u_{30}$ $u_{31}$ $u_{22}$ $u_{23}$ $u_{32}$ $u_{33}$ $H^{u_{0}}$ $H^{u_{1}}$
Figure 3: The graph $G_{1}$
$u_{0}$ $u_{1}$ $u_{2}$ $u_{3}$
Figure 4: The graph $G_{2}$
### 2.2 A transitional theorem and its proof
In order to prove our main result (Theorem 1.5), we first prove the following transitional theorem by constructing $\ell-\left\lceil k/2\right\rceil$ internally disjoint $U$ -Steiner trees in $G$ .
**Theorem 2.1**
*Let $G=S(n,\ell)$ and $U\subseteq V(G)$ with $|U|=k$ ( $U$ is defined before), and let $c=\ell-\left\lceil k/2\right\rceil$ . For $0\leq s\leq n$ , $G_{s}$ has $c$ internally disjoint $U^{s}$ -Steiner trees $T_{1},\cdots,T_{c}$ such that the following statement holds.
1. For each $u\in U^{s}$ (say $u\in W^{v}$ and hence $v$ is a labelled vertex of $G_{s+1}$ if $s\neq n$ ) and $i\in[c]$ , $d_{T_{i}}(u)\leq 2$ .*
The rest part of this subsection is the proof of Theorem 2.1.
Proof of Theorem 2.1 Suppose that $s^{\prime}$ is the minimum integer such that $G_{s^{\prime}}$ has exactly one labelled vertex (note that $s^{\prime}$ exists since $G_{n}$ consists of a single labelled vertex). This means that $G_{s^{\prime}-1}$ has at least two labelled vertices. Let $v$ be the labelled vertex in $G_{s^{\prime}}$ . Then $U\subseteq A_{s^{\prime},v}$ . Thus, in order to find $c$ internally disjoint $U$ -Steiner trees of $G$ satisfying ( $\star$ ), we only need to find $c$ internally disjoint $U$ -Steiner trees in $A_{s^{\prime},v}$ satisfying ( $\star$ ). Hence, without loss of generality, we can assume that $G$ is a graph with
$$
|W^{v_{root}}|=|U^{n-1}|\geq 2. \tag{1}
$$
Therefore, $|U^{i}|\geq 2$ for each $i\leq n-1$ . $u_{11}$ $u_{12}$ $u_{13}$ $u_{24}$ $u_{21}$ $u_{22}$ $u_{23}$ $u_{24}$ $u_{31}$ $u_{32}$ $u_{41}$ $u_{42}$ $u_{33}$ $u_{34}$ $u_{43}$ $u_{44}$ $e$ $u_{e}$ $v_{e}$ $u$ $v$ $e$ $G_{1}$ $G_{2}$
Figure 5: From graph $G_{1}$ to $G_{2}$ by $u_{e}v_{e}$ contraction edge $uv$ .
The proof is technical and is via induction. Note that since $G_{n}$ is a single vertex and $G=G_{0}$ , the induction is from $G_{n}$ to $G_{0}$ . The basic idea is to use the $U^{s}$ -Steiner trees of $G_{s}$ to construct appropriate $U^{s-1}$ -Steiner trees of $G_{s-1}$ . Since each vertex in $G_{s}$ corresponds to a complete graph in $G_{s-1}$ , it is not a straightforward process to extend $U^{s}$ -Steiner trees of $G_{s}$ to $U^{s-1}$ -Steiner trees of $G_{s-1}$ .
If $s=n$ , then each $T_{i}$ in $G_{n}$ is the empty graph and the result holds. Thus, suppose $s\leq n-1$ . Hence, the labelled vertex $v$ of $G_{s+1}$ always exists. The following implies that the result holds for $s=n-1$ . Note that $U^{n-1}=W^{v_{root}}$ .
**Claim 1**
*If $s=n-1$ , then we can construct $c$ internally disjoint $U^{s}$ -Steiner trees, say $T_{1},\ldots,T_{c}$ , such that for each $i\in[c]$ and $u\in U^{s}$ , $d_{T_{i}}(u)\leq 2$ . Moreover,
1. $(i)$ If $|U^{s}|\leq\lceil k/2\rceil$ , then for each $i\in[c]$ and $u\in U^{s}$ , $d_{T_{i}}(u)=1$ ;
1. $(ii)$ otherwise, for each $u\in U^{s}$ , there are at most $|U^{s}|-\lceil k/2\rceil$ internally disjoint $U^{s}$ -Steiner trees $T_{i}$ such that $d_{T_{i}}(u)=2$ .*
* Proof*
Note that $G_{n-1}=S(1,\ell)=K_{\ell}$ . Suppose $|U^{n-1}|\leq\left\lceil k/2\right\rceil$ . Then $|V(G_{n-1})-U^{n-1}|\geq\ell-\left\lceil k/2\right\rceil=c$ and we can choose $c$ stars of $\{x\vee U^{n-1}:\ x\in V(G_{n-1})-U^{n-1}\}$ as internally disjoint $U^{n-1}$ -Steiner trees $T_{1},\ldots,T_{c}$ . It is easy to verify that $d_{T_{i}}(u)=1$ for $i\in[c]$ and $u\in U^{n-1}$ . Suppose $|U^{n-1}|\geq\lceil k/2\rceil$ . Since $|U^{n-1}|\leq k$ , it follows that $0\leq|U^{n-1}|-\left\lceil k/2\right\rceil\leq\lfloor|U^{n-1}|/2\rfloor$ . By Corollary 1.4, there are $|U^{n-1}|-\left\lceil k/2\right\rceil$ edge-disjoint Hamiltonian paths in $G_{n-1}[U^{n-1}]$ . So, we can choose $c$ internally disjoint $U^{n-1}$ -Steiner trees consisting of $|U^{n-1}|-\left\lceil k/2\right\rceil$ edge-disjoint Hamiltonian paths in $G_{n-1}[U^{n-1}]$ and $\ell-|U^{n-1}|$ stars $\{x\vee U^{n-1}:\ x\in V(G_{n-1})-U^{n-1}\}$ . It is easy to verify that $d_{T_{i}}(u)\leq 2$ for $i\in[c]$ and $u\in U^{n-1}$ , and there are at most $|U^{n-1}|-\left\lceil k/2\right\rceil$ internally disjoint $U^{n-1}$ -Steiner trees $T_{i}$ such that $d_{T_{i}}(u)=2$ . ∎
Our proof is a recursive process that the $c$ internally disjoint $U^{s}$ -Steiner trees of $G_{s}$ are constructed by using the $c$ internally disjoint $U^{s+1}$ -Steiner trees of $G_{s+1}$ . We will find some ways to construct the $c$ internally disjoint $U^{s}$ -Steiner trees such that $(\star)$ holds. In the finial step, the $c$ internally disjoint $U$ -Steiner trees in $G_{0}=G$ will be obtained. We have proved that the result holds for $s\in\{n,n-1\}$ . Now, suppose that we have constructed $c$ internally disjoint $U^{s+1}$ -Steiner trees, say $\mathcal{F}=\{F_{1},\ldots,F_{c}\}$ , and the trees satisfy ( $\star$ ), where $s\in\{0,\ldots,n-1\}$ . We need to construct $c$ internally disjoint $U^{s}$ -Steiner trees $\mathcal{T}=\{T_{1},\ldots,T_{c}\}$ of $G_{s}$ satisfying ( $\star$ ).
Recall that each edge of $G_{s+1}$ is also an edge of $G_{s}$ . In addition, let $E_{u,i}$ denote the set of edges in $F_{i}$ incident with the labelled vertex $u$ in $G_{s}$ and let $V_{u,i}=\{u_{e}:\ e\in E_{u,i}\}$ (recall that the edge $e=uw$ in $G_{s+1}$ is denoted by $e=u_{e}w_{e}$ in $G_{s}$ , where $u_{e}\in V(H^{u})$ and $w_{e}\in V(H^{w})$ ). Note that $E_{u,i}$ and $V_{u,i}$ are the subsets of $E(G_{s})$ and $V(G_{s})$ , respectively.
Our aim is to construct internally disjoint $U^{s}$ -Steiner trees $T_{1},\ldots,T_{c}$ that is obtained from $U^{s+1}$ -Steiner trees $F_{1},\ldots,F_{c}$ . So, we need to “blow” up each vertex $w$ of $G_{s+1}$ into $H^{w}$ and find internally disjoint $W^{w}$ -Steiner trees $F_{1}^{w},\ldots,F_{c}^{w}$ of $H^{w}$ ( $F_{i}^{w}$ may be the empty graph if $w$ is a unlabelled vertex) such that
$$
\mathcal{T}=\left\{T_{i}=F_{i}\cup\bigcup_{w\in V(G_{s+1})}F_{i}^{w}:\ i\in[c]
\right\}.
$$
Each $F_{i}^{w}$ can be constructed as follows.
- If $w$ is a labelled vertex of $G_{s+1}$ , then $w$ is in each $U^{s+1}$ -Steiner tree of $\mathcal{F}$ . We choose $c$ internally disjoint $W^{w}$ -Steiner trees $F_{1}^{w},\ldots,F_{c}^{w}$ of $H^{w}$ such that $V_{w,i}\subseteq V(F_{i}^{w})$ for each $i\in[c]$ .
- If $w$ is an unlabelled vertex, then $w$ is in at most one $U^{s+1}$ -Steiner tree of $\mathcal{F}$ . For each $i\in[c]$ , if $w\in V(F_{i})$ , then let $F_{i}^{w}$ be a spanning tree of $H^{w}$ ; if $w\notin V(F_{i})$ , then let $F_{i}^{w}$ be the empty graph.
For $i\in[c]$ , let $E_{i}=E(F_{i})\cup\bigcup_{w\in V(G_{s+1})}E(F_{i}^{w})$ and let $T_{i}=G_{s}[E_{i}]$ . It is obvious that $T_{1},\ldots,T_{c}$ are internally disjoint $U^{s}$ -Steiner trees. We need to ensure that each labelled vertex of $G^{s}$ satisfies ( $\star$ ).
In fact, without loss of generality, we only need to choose an arbitrary labelled vertex $u\in G_{s+1}$ and prove that each vertex of $W^{u}$ satisfies ( $\star$ ). This is because $F_{i}^{w}$ is clear when $w$ is an unlabelled vertex (in the case $F_{i}^{w}$ is either a spanning tree of $H^{w}$ or the empty graph). By Eq. (1) and Fact 1, $|W^{u}|\leq k-1$ since $u\neq v_{root}$ .
Since $d_{F_{i}}(u)\leq 2$ for each $i\in[c]$ and $u\in U^{s}$ , it follows that $|E_{u,i}|=|V_{u,i}|\leq 2$ . Therefore, we can divide $F_{1},\ldots,F_{c}$ into the following five types on $u$ :
Type 1: $|V_{u,i}|=1$ and $V_{u,i}\subseteq W^{u}$ .
Type 2: $|V_{u,i}|=1$ and $V_{u,i}\nsubseteq W^{u}$ .
Type 3: $|V_{u,i}|=2$ and $V_{u,i}\subseteq W^{u}$ .
Type 4: $|V_{u,i}|=2$ and $V_{u,i}\cap W^{u}=\emptyset$ .
Type 5: $|V_{u,i}|=2$ and $|V_{u,i}\cap W^{u}|=1$ .
If $F_{i}$ is a graph of Type $j\ (1\leq j\leq 5)$ on $u$ , then we also call $F_{i}^{u}$ a $W^{u}$ -Steiner tree of Type $j$ . Suppose there are $n_{j}(u)$ trees $F_{i}$ that belong to Type $j$ on $u$ for $j\in[5]$ . Then
$$
\displaystyle\sum_{i=1}^{5}n_{j}(u)=c. \tag{2}
$$
It is obvious that $n_{j}(v_{root})=0$ for each $j\in[5]$ . Let
$$
R(u)=V(H^{u})-W^{u}-\bigcup_{i\in[c]}V_{u,i}.
$$
Note that $R(u)$ is the set of unlabelled vertices in $G_{s}$ , which will be used to construct the $W^{u}$ -Steiner trees in $H^{u}$ . It is clear that
$$
|R(u)|=\ell-|W^{u}|-[n_{2}(u)+2n_{4}(u)+n_{5}(u)]. \tag{3}
$$
<details>
<summary>x2.png Details</summary>
### Visual Description
## Diagram: Configurations of W^u and H^u Across Methods and Types
### Overview
The image presents a technical diagram illustrating five distinct configurations (Type 1–Type 5) of two sets, **W^u** (upper set) and **H^u** (lower set), connected by lines. Below these, three methods (Method 1, Method 2, Method 3) and a sub-method (Method 1a) are depicted, each combining specific types with unique line arrangements. The diagram uses black lines, dotted lines, and labeled points (e.g., **x**, **y**) to represent relationships.
---
### Components/Axes
1. **Primary Diagrams (Top Row)**:
- **Type 1**: Single point on **W^u** (top) and **H^u** (bottom), connected by a vertical line.
- **Type 2**: Two points on **W^u**, connected by a horizontal line; one point on **H^u** below.
- **Type 3**: Two points on **W^u** (top) and one on **H^u** (bottom), with lines connecting **W^u** points to **H^u**.
- **Type 4**: Two points on **W^u** and two on **H^u**, with cross-connections between **W^u** and **H^u**.
- **Type 5**: One point on **W^u** and two on **H^u**, with lines connecting **W^u** to both **H^u** points.
2. **Method Boxes (Bottom Section)**:
- **Method 1**: Combines Type 1 (left) and Type 3 (right), with solid lines.
- **Method 1a**: Similar to Method 1 but uses dotted lines for connections.
- **Method 2**: Combines Type 2 (left) and Type 5 (right), with solid lines.
- **Method 3**: Single diagram (Type 4) with a new point **y** connected to **x** via a dashed line.
3. **Labels**:
- **W^u**: Consistently placed at the top of all diagrams.
- **H^u**: Positioned at the bottom of all diagrams.
- **x**, **y**: Points labeled in specific configurations (e.g., Method 3, Type 4).
---
### Detailed Analysis
- **Type 1**: Minimal configuration with one point per set. Vertical alignment suggests a direct relationship.
- **Type 2**: Two **W^u** points imply a binary interaction; **H^u** acts as a terminal node.
- **Type 3**: Introduces multiple **W^u** points connected to a single **H^u**, indicating a many-to-one relationship.
- **Type 4**: Complex cross-connections between **W^u** and **H^u** points, suggesting a networked structure.
- **Type 5**: Reverses the hierarchy, with **H^u** dominating (**H^u** has two points vs. one **W^u**).
- **Method 1a**: Dotted lines in Type 1 and Type 3 may represent hypothetical or alternative pathways.
- **Method 3**: Introduces **y** as a new variable, connected to **x** via a dashed line, possibly denoting an indirect or secondary relationship.
---
### Key Observations
1. **Hierarchical Variation**: Types 1–5 show increasing complexity, from simple vertical connections (Type 1) to networked structures (Type 4/5).
2. **Method Differentiation**:
- **Method 1a** (dotted lines) contrasts with **Method 1** (solid lines), possibly indicating uncertainty or alternative hypotheses.
- **Method 3** introduces **y**, expanding the system beyond previous configurations.
3. **Spatial Consistency**: **W^u** and **H^u** are always vertically aligned, reinforcing their roles as distinct sets.
---
### Interpretation
The diagram likely models a system where **W^u** and **H^u** represent distinct entities (e.g., variables, states, or components) with varying interaction patterns. The methods suggest different analytical or operational approaches:
- **Method 1/1a**: Focus on foundational or alternative configurations.
- **Method 2**: Emphasizes binary and reversed-hierarchy interactions.
- **Method 3**: Introduces a new variable (**y**), hinting at expanded scope or emergent behavior.
The use of dotted/dashed lines in **Method 1a** and **Method 3** may signal provisional or secondary relationships, requiring further validation. The progression from Type 1 to Type 5 reflects a spectrum of complexity, potentially mapping stages of a process or scenarios in a decision-making framework.
</details>
Figure 6: Five types and methods 1, 1a, 2 and 3.
We have the following five methods to choose $F_{1}^{u},\ldots,F_{c}^{u}$ for the labelled vertex $u\in G_{s+1}$ (see Figure 6).
Method 1 If $F_{i}$ is a graph of Type 1 and $|W^{u}|\geq 2$ , or $F_{i}$ is a graph of Type 3, then let $F^{u}_{i}=x\vee W^{u}$ , where $x\in R(u)$ .
Method 1a If $F_{i}$ is a graph of Type 1 and $|W^{u}|\geq 2$ , then let $F^{u}_{i}$ be a Hamiltonian path of $H^{u}[W^{u}]$ such that the vertex in $V_{u,i}$ is an endpoint of this Hamiltonian path; if $F_{i}$ is a graph of Type 3, then let $F^{u}_{i}$ be a Hamiltonian path of $H^{u}[W^{u}]$ such that the endpoints of $F_{i}$ are two vertices in $V_{u,i}$ .
Method 2 If $F_{i}$ is a graph of Type 2 or Type 5, say $x$ is the only vertex of $V_{u,i}$ with $x\notin W^{u}$ , then let $F^{u}_{i}=x\vee W^{u}$ .
Method 3 If $F_{i}$ is a graph of Type 4, say $V_{u,i}=\{x,y\}$ , then let $F^{u}_{i}=xy\cup(x\vee W^{u})$ .
Method 4 If $F_{i}$ is a graph of Type 1 and $|W^{u}|=1$ , then let $F_{i}^{u}$ be the empty graph.
It is worth noting that the method is deterministic if $F_{i}^{u}$ is a graph of Types 2, 4 and 5, or $F_{i}^{u}$ is a graph of Type 1 and $|W^{u}|=1$ . So we firstly construct these $F_{i}^{u}$ s by using Methods 2, 3 and 4, respectively, and then construct $F^{u}_{i}$ s of Type 1 with $|W^{u}|\geq 2$ and construct $F^{u}_{i}$ s of Type 3 by using Method 1 or Method 1a.
The following is an algorithm constructing $c$ internally disjoint $U$ -Steiner trees of $G$ .
Input: $U$ , $c$ and $G=S(n,\ell)$ with $|W^{root}|\geq 2$
Output: $c$ internally disjoint $U$ -Steiner trees $T^{\prime}_{1},T^{\prime}_{2},\ldots,T^{\prime}_{c}$ of $G$
1 $T^{\prime}_{1},T^{\prime}_{2},\ldots,T^{\prime}_{c}$ are empty graphs;
2 $s=n$ //(* in initial step, $G^{n}$ is a single vertex*);
3 for $s\geq 1$ do
4 for each vertex $u$ of $G_{s}$ do
5 if $u$ is an unlabelled vertex then
6 for $1\leq i\leq c$ do
7 if $u$ is a vertex of $F_{i}$ then
8 $F_{i}^{u}$ is a spanning tree of $H^{u}$ ;
9 else
10 $\ F_{i}^{u}$ is is the empty graph;
11 end if
12 $T^{\prime}_{i}=T^{\prime}_{i}\cup F_{i}^{u}$ ;
13 $i=i+1$ ;
14
15 end for
16
17 end if
18
19 if $u$ is a labelled vertex then
20 $\mu=|R(u)|$ ;
21 for $1\leq i\leq c$ do
22
23 if $T^{\prime}_{i}$ is of Type 2 or Type 5 then
24 construct $F_{i}^{u}$ by using Method 2;
25
26 end if
27
28 if $T^{\prime}_{i}$ is of Type 4 then
29 construct $F_{i}^{u}$ by using Method 3;
30
31 end if
32
33 if $T^{\prime}_{i}$ is of Type 1 and $|W^{u}|=1$ then
34 construct $F_{i}^{u}$ by using Method 4;
35
36 end if
37
38 if either $T^{\prime}_{i}$ is of Type 1 and $|W^{u}|\geq 2$ , or $T^{\prime}_{i}$ is of Type 3 then
39 if $\mu>0$ then
40 construct $F_{i}^{u}$ by using Method 1;
41 $\mu=\mu-1$ ;
42
43 end if
44
45 if $\mu\leq 0$ then
46 construct $F_{i}^{u}$ by using Method 1a;
47
48 end if
49
50 end if
51
52 $T^{\prime}_{i}=T^{\prime}_{i}\cup F_{i}^{u}$ ;
53 $i=i+1$ ;
54
55 end for
56
57 end if
58
59 end for
60 $s=s-1$ ;
61
62 end for
Algorithm 1 The construction of $c$ internally disjoint $U$ -Steiner trees of $G$
Algorithm 1 is an algorithm for finding $c$ internally disjoint $U$ -Steiner trees of $G$ . For each step of $s$ , the algorithm will construct $c$ internally disjoint $U^{s-1}$ -steiner trees $T^{\prime}_{1},T^{\prime}_{2},\ldots,T^{\prime}_{c}$ . For convenience, we denote each $T_{i}^{\prime}$ by $T_{i}^{s}$ after the $s$ step of outer “for”, that is, $T_{1}^{s},T_{2}^{s},\ldots,T_{c}^{s}$ are $c$ internally disjoint $U^{s}$ -Steiner trees in $G^{s}$ generated in Algorithm 1. If Algorithm 1 is correct, then Lines 16–40 indicate that $d_{T^{\prime}_{i}}(x)\leq 2$ for any $x\in W^{u}$ , and hence ( $\star$ ) always holds.
We now check the correctness of the algorithm. Since $F_{i}^{u}$ is deterministic when $u$ is an unlabelled vertex (Lines 5–15 of Algorithm 1), and $F_{i}^{u}$ is deterministic if $u$ is a labelled vertex and either $F_{i}^{u}$ is a graph of Types 2, 4 and 5, or $F_{i}^{u}$ is a graph of Type 1 and $|W^{u}|=1$ (Lines 19–27 of Algorithm 1), we only need to talk about the labelled vertex $u$ with $|W^{u}|\geq 2$ and check the correctness of Lines 28–36 of Algorithm 1. That is, to ensure that there exist $n_{1}(u)$ $F_{i}^{u}$ s of Type 1 and $n_{3}(u)$ $F_{i}^{u}$ s of Type 3. If $F_{i}^{u}$ is constructed by Method 1, then $F_{i}^{u}$ is a star with center in $R(u)$ (say the center of the star is $a_{i}$ ); if $F_{i}^{u}$ is constructed by Method 1a, then $F_{i}^{u}$ is a Hamiltonian path of $H^{u}$ with endpoints in $V_{u,i}$ . Since $a_{i}$ s are pairwise differently and are contained in $R(u)$ , and $H^{u}$ has at most $\lfloor|W^{u}|/2\rfloor$ Hamiltonian paths to afford by Corollary 1.4, we only need to ensure that $|R(u)|+\lfloor|W^{u}|/2\rfloor\geq n_{1}(u)+n_{3}(u)$ . Thus, we only need to prove the following result.
**Lemma 2.1**
*For each labelled vertex $u\in V(G_{s})$ with $|W^{u}|\geq 2$ , Ineq.
$$
\displaystyle n_{1}(u)+n_{3}(u)-|R(u)|\leq\lfloor|W^{u}|/2\rfloor \tag{4}
$$
holds.*
* Proof*
By Eqs. (2) and (3),
| | $\displaystyle n_{1}(u)+n_{3}(u)-|R(u)|$ | $\displaystyle=n_{1}(u)+n_{3}(u)-[\ell-|W^{u}|-n_{2}(u)-2n_{4}(u)-n_{5}(u)]$ | |
| --- | --- | --- | --- |
Since $c=\ell-\lceil k/2\rceil$ , it follows that
$$
\displaystyle n_{1}(u)+n_{3}(u)-|R(u)|=n_{4}(u)+|W^{u}|-\lceil k/2\rceil. \tag{5}
$$ Before the proof of Lemma 2.1, we give a series of claims as preliminaries.
**Claim 2**
*Suppose that $a\in V(v_{root}\overrightarrow{T}u)$ is a labelled vertex with $a\in U^{\iota}$ , where $1\leq\iota\leq n$ . If $|W^{a}|=1$ (say $W^{a}=\{x\}$ ), then $n_{5}(a)\leq 1$ . Moreover,
1. if $n_{5}(a)=1$ , then $n_{4}(x)\leq 1$ ;
1. if $n_{5}(a)=0$ , then $n_{3}(x)=n_{4}(x)=n_{5}(x)=0$ .*
* Proof*
Note that $x\in U^{\iota-1}$ . in Algorithm 1 (Lines 18–39), if $T_{i}^{\iota}$ is not a tree of Type 5 on $a$ , then $F_{i}^{a}$ is chosen such that $d_{T_{i}^{\iota-1}}(x)=1$ (here $W^{a}=\{x\}$ ); if $T^{\iota}_{i}$ is a tree of Type 5 on $a$ , then $F_{i}^{a}$ is chosen such that $d_{T_{i}^{\iota-1}}(x)=2$ . Since $|W^{a}|=1$ , there is at most one $T^{\iota}_{i}$ of Type 5 on $a$ , and hence $n_{5}(a)\leq 1$ . Furthermore, if $n_{5}(a)=1$ , then there is exact one $T_{i}^{\iota-1}$ in $G^{\ell-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ . Hence, $n_{4}(x)\leq 1$ . If $n_{5}(a)=0$ , then $d_{T_{i}^{\iota-1}}(x)=1$ for each $T_{i}^{\iota-1}$ . Therefore, $n_{3}(x)=n_{4}(x)=n_{5}(x)=0$ . ∎
**Claim 3**
*Suppose that $a,b$ are two labelled vertices of $v_{root}\overrightarrow{T}u$ and $a\in W^{b}$ , where $a\in U^{\iota}$ for some $\iota\in[n]$ . Then there are at most $\max\{1,n_{1}(b)+n_{3}(b)-|R(b)|+1\}$ $T_{i}^{\iota}$ s such that $d_{T_{i}^{\iota}}(a)=2$ . Furthermore, $n_{4}(a)\leq\max\{1,n_{1}(b)+n_{3}(b)-|R(b)|+1\}$ .*
* Proof*
Since $a\in W^{b}$ and $a\in U^{\iota}$ , it follows that $b\in U^{\iota+1}$ . For an $i\in[c]$ with $d_{T_{i}^{\iota}}(a)=2$ , we have that either $F_{i}^{b}$ is a Hamiltonian path of the clique $H^{b}[W^{b}]$ , or $a\in V_{b,i}\cap W^{b}$ for some $i\in[c]$ (recall that $V_{b,i}=\{z\in V(H^{b}):z\mbox{ is an end vertex of some edge in }E_{b,i}\}$ , where $E_{b,i}$ is the set of edges in $T_{i}^{\iota+1}$ incident with $b$ ). Hence, if there are $h$ edge-disjoint $F_{i}^{b}$ s that are constructed as Hamiltonian paths of $H^{b}[W^{b}]$ , then there are at most $h+1$ internally disjoint $U^{\iota}$ -Steiner trees $T_{i}^{\iota}$ such that $d_{T_{i}^{\iota}}(a)=2$ . By the definition of $n_{4}(a)$ , we have that $n_{4}(a)\leq h+1$ . In Algorithm 1 (Lines 28–36), if $n_{1}(b)+n_{3}(b)-|R(b)|>0$ , then there are at most $n_{1}(b)+n_{3}(b)-|R(b)|$ $F_{i}^{b}$ s that are constructed as Hamiltonian paths in $H^{b}[W^{b}]$ ; if $n_{1}(b)+n_{3}(b)-|R(b)|\leq 0$ , there is no $F_{i}^{b}$ that is constructed as a Hamiltonian path in $H^{b}[W^{b}]$ . Hence, $h\leq\max\{0,n_{1}(b)+n_{3}(b)-|R(b)|\}$ . Thus, there are at most $\max\{1,n_{1}(b)+n_{3}(b)-|R(b)|+1\}$ internally disjoint $U^{\iota}$ -Steiner trees $T_{i}^{\iota}$ such that $d_{T_{i}^{\iota}}(a)=2$ , and $n_{4}(a)\leq\max\{1,n_{1}(b)+n_{3}(b)-|R(b)|+1\}$ . ∎
**Claim 4**
*Let $a\in V(v_{root}\overrightarrow{T}u)$ be a labelled vertex, where $a\in U^{\iota}$ for some $\iota\in[n]$ . If $n_{4}(a)\leq 1$ and $2\leq|W^{a}|\leq\left\lceil k/2\right\rceil-1$ , then $n_{1}(a)+n_{3}(a)-|R(a)|\leq 0$ . Moreover, for each vertex $x\in W^{a}$ , there is at most one $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ .*
* Proof*
Since $n_{4}(a)\leq 1$ and $|W^{a}|\leq\left\lceil k/2\right\rceil-1$ , it follows from Eq. (5) that $n_{1}(a)+n_{3}(a)-|R(a)|=|W^{a}|-\left\lceil k/2\right\rceil+1\leq 0.$ By Claim 3, for each vertex $x\in W^{a}$ , there is at most one $T_{i}^{a}$ such that $d_{T_{i}^{\iota-1}}(x)\leq 2$ . ∎
**Claim 5**
*Let $a\in V(v_{root}\overrightarrow{T}u)$ be a labelled vertex, where $a\in U^{\iota}$ for some $\iota\in[n]$ . If $n_{4}(a)\leq 1$ and $\left\lceil k/2\right\rceil\leq|W^{a}|\leq k-1$ , then $n_{1}(a)+n_{3}(a)-|R(a)|\leq\lfloor|W^{a}|/2\rfloor$ . Moreover, for each $x\in W^{a}$ ,
1. if $n_{4}(a)=1$ , then there are at most $|W^{a}|-\left\lceil k/2\right\rceil+2$ internally disjoint $U^{\iota-1}$ -Steiner trees $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ ;
1. if $n_{4}(a)=0$ , then there are at most $|W^{a}|-\left\lceil k/2\right\rceil+1$ internally disjoint $U^{\iota-1}$ -Steiner trees $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ .*
* Proof*
Since $n_{4}(a)\leq 1$ , it follows from Eq. (5) that
| | $\displaystyle n_{1}(a)+n_{3}(a)-|R(a)|$ | $\displaystyle\leq|W^{a}|-\left\lceil k/2\right\rceil+n_{4}(a)$ | |
| --- | --- | --- | --- |
and the equality indicates $n_{4}(a)=1$ . If $|W^{a}|\leq k-2$ , then
$$
n_{1}(a)+n_{3}(a)-|R(a)|\leq|W^{a}|-\left\lceil(|W^{a}|+2)/2\right\rceil+1\leq
\left\lfloor|W^{a}|/2\right\rfloor;
$$
if $|W^{a}|=k-1$ and $k$ is even, then
$$
n_{1}(a)+n_{3}(a)-|R(a)|=|W^{a}|-\left\lceil(|W^{a}|+1)/2\right\rceil+1=\left
\lfloor(|W^{a}|+1)/2\right\rfloor=\left\lfloor|W^{a}|/2\right\rfloor;
$$
if $|W^{a}|=k-1$ and $k$ is odd, then
$$
n_{1}(a)+n_{3}(a)-|R(a)|=(k-1)-\left\lceil k/2\right\rceil+1\leq\left\lfloor k
/2\right\rfloor=\left\lfloor(k-1)/2\right\rfloor=\left\lfloor|W^{a}|/2\right\rfloor.
$$
Therefore, $n_{1}(a)+n_{3}(a)-|R(a)|\leq\left\lfloor|W^{a}|/2\right\rfloor$ . By Eq. (5) and Claim 3, if $n_{4}(a)=1$ , then for each vertex $x\in W^{a}$ , there are at most $n_{1}(a)+n_{3}(a)-|R(a)|+1=|W^{a}|-\left\lceil k/2\right\rceil+2$ internally disjoint $U^{\iota-1}$ -Steiner trees $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ ; if $n_{4}(a)=0$ , then for each vertex $x\in W^{a}$ , there are at most $n_{1}(a)+n_{3}(a)-|R(a)|+1=|W^{u}|-\left\lceil k/2\right\rceil+1$ internally disjoint $U^{\iota-1}$ -Steiner trees $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ . ∎
**Claim 6**
*Suppose $a,b\in V(\overrightarrow{T})$ , $a\prec b$ and $a\overrightarrow{T}b=az_{1}z_{2}\ldots z_{p}b$ , where $a\in U^{\iota}$ for some $\iota\in[n]$ . If $|W^{z_{i}}|\leq\lceil k/2\rceil-1$ for each $i\in[p]$ , and either $|W^{a}|=1$ or $2\leq|W^{a}|\leq\lceil k/2\rceil-1$ and $n_{4}(a)\leq 1$ , then $n_{4}(b)\leq 1$ .*
* Proof*
Since $a\in U^{\iota}$ , it follows that $z_{q}\in U^{\iota-q}$ for each $q\in[p]$ and $b\in U^{\iota-p-1}$ . Since $|W^{a}|=1$ or $2\leq|W^{a}|\leq\lceil k/2\rceil-1$ and $n_{4}(a)\leq 1$ , by Claims 2 and 4, there are at most one $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(z_{1})=2$ . Hence, $n_{4}(z_{1})\leq 1$ . Since $|W^{z_{1}}|\leq\lceil k/2\rceil-1$ and $n_{4}(z_{1})\leq 1$ , by Claims 2 and 4, there are at most one $T_{i}^{\iota-2}$ such that $d_{T_{i}^{\iota-2}}(z_{2})=2$ . Hence, $n_{4}(z_{2})\leq 1$ . Repeat this progress, we can get that $n_{4}(z_{p})\leq 1$ . Since $|W^{z_{p}}|\leq\lceil k/2\rceil-1$ and $n_{4}(z_{p})\leq 1$ , by Claims 2 and 4, there are at most one $T_{i}^{\iota-p-1}$ such that $d_{T_{i}^{\iota-p-1}}(b)=2$ . Hence, $n_{4}(b)\leq 1$ . ∎
**Claim 7**
*Suppose $|W^{v_{root}}|\leq\lceil k/2\rceil$ , $b\in V(\overrightarrow{T})$ and $v_{root}\overrightarrow{T}b=v_{root}z_{1}z_{2}\ldots z_{p}b$ . If $|W^{z_{i}}|=1$ for each $i\in[p]$ , then $n_{4}(b)=0$ .*
* Proof*
Since $|W^{v_{root}}|\leq\lceil k/2\rceil$ , by Claim 1, $d_{T_{i}^{n-1}}(z_{1})=1$ for each $U^{n-1}$ -Steiner tree $T_{i}^{n-1}$ . Hence, $n_{3}(z_{1})=n_{4}(z_{1})=n_{5}(z_{1})=0$ . Since $|W^{z_{1}}|=1$ and $n_{5}(z_{1})=0$ , by the second statement of Claim 2, $n_{3}(z_{2})=n_{4}(z_{2})=n_{5}(z_{2})=0$ . Since $|W^{z_{2}}|=1$ and $n_{5}(z_{2})=0$ , by the second statement of Claim 2, $n_{3}(z_{3})=n_{4}(z_{3})=n_{5}(z_{3})=0$ . Repeat this process, we get that $n_{4}(b)=0$ . ∎
**Claim 8**
*Suppose that $a,b$ are two labelled vertices and $a\in W^{b}$ , where $a\in U^{\iota}$ for some $\iota\in[n]$ . If $k$ is even and $|W^{b}|=|W^{a}|=k/2$ , then $n_{1}(a)+n_{3}(a)-|R(a)|\leq\lfloor|W^{a}|/2\rfloor$ and the following hold.
1. If $b=v_{root}$ , then for each vertex $x\in W^{a}$ , there is at most one $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ .
1. If $b\neq v_{root}$ , then for each vertex $x\in W^{a}$ , there are at most two $T_{i}^{\iota-1}$ s such that $d_{T_{i}^{\iota-1}}(x)=2$ .*
* Proof*
Suppose $b=v_{root}$ . Then by Claim 1, $n_{4}(a)=0$ . Thus,
$$
n_{1}(a)+n_{3}(a)-|R(a)|=n_{4}(a)+|W^{a}|-k/2=|W^{a}|-k/2=0.
$$
By Claim 3, for each vertex $x\in W^{a}$ , there is at most one $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ . Now assume that $b\neq v_{root}$ . Without loss of generality, suppose $v_{root}\overrightarrow{T}b=v_{root}v_{1}v_{2}\ldots v_{p}b$ . By Fact 1, we have that $|W^{root}|=2\leq k/2$ and $|W^{v_{i}}|=1$ for each $i\in[p]$ . By Claim 7, $n_{4}(b)=0$ . By Claim 3,
| | $\displaystyle n_{4}(a)$ | $\displaystyle\leq\max\{1,n_{1}(b)+n_{3}(b)-|R(b)|+1\}$ | |
| --- | --- | --- | --- |
By Claim 3 again, for each vertex $x\in W^{a}$ , there are at most two $T_{i}^{\iota-1}$ such that $d_{T_{i}^{\iota-1}}(x)=2$ . ∎
With the above preparations, we now prove Lemma 2.1. Recall that $u\in V(G_{s})$ is a labelled vertex with $|W^{u}|\geq 2$ . Then each vertex of $V(v_{root}\overrightarrow{T}u)$ is also a labelled vertex. Suppose that $v^{*}$ is the maximum vertex of $v_{root}\overrightarrow{T}u$ such that one of the following holds (if such vertex $v^{*}$ exists).
1. $|W^{v^{*}}|=1$ ,
1. $2\leq|W^{v^{*}}|\leq\lceil k/2\rceil-1$ and $n_{4}(v^{*})\leq 1$ .
We distinguish the following two cases to show this lemma, that is, to prove Ineq. (4) holds (recall that the Ineq. (4) is $n_{1}(u)+n_{3}(u)-|R(u)|\leq\lfloor|W^{u}|/2\rfloor$ ).
**Case 1**
*$v^{*}$ exists.*
Let $\overrightarrow{P}=v^{*}\overrightarrow{T}u=v^{*}z_{1}z_{2}\ldots z_{p}u$ . By the maximality of $v^{*}$ , we have that $|W^{z_{i}}|\geq 2$ for each $i\in[p]$ . If $|\overrightarrow{P}|=1$ , then $v^{*}=u$ . Since $|W^{u}|\geq 2$ , it follows from ( $ii$ ) that $2\leq|W^{u}|\leq\lceil k/2\rceil-1$ and $n_{4}(u)\leq 1$ . By Claim 4, we have that $n_{1}(u)+n_{3}(u)-|R(u)|\leq 0$ , and Ineq. (4) holds. Thus, we assume that $|\overrightarrow{P}|\geq 2$ below. By Claim 6, we have that
$$
\displaystyle n_{4}(z_{1})\leq 1. \tag{6}
$$
Thus, by the maximality of $v^{*}$ , we have that $|W^{z_{1}}|\geq\lceil k/2\rceil$ . Since $|W^{v_{root}}|+|W^{z_{1}}|\leq k+1$ (by Fact 1) and $|W^{v_{root}}|\geq 2$ , it follows that $|W^{z_{1}}|\leq k-1$ . Hence,
$$
\displaystyle\lceil k/2\rceil\leq|W^{z_{1}}|\leq k-1. \tag{7}
$$
**Subcase 1.1**
*$p=0$ .*
Then $\overrightarrow{P}=v^{*}\overrightarrow{T}u=v^{*}u$ and $z_{1}=u$ . Since $n_{4}(u)\leq 1$ and $\lceil k/2\rceil\leq|W^{u}|\leq k-1$ by Ineqs. (6) and (7), it follows from Claim 5 that $n_{1}(u)+n_{3}(u)-|R(u)|\leq\lfloor|W^{u}|/2\rfloor$ , Ineq. (4) holds.
**Subcase 1.2**
*$p=1$ .*
Then $\overrightarrow{P}=v^{*}\overrightarrow{T}u=v^{*}z_{1}u$ and $u=z_{2}$ . By Fact 1, we have that
$$
\displaystyle|W^{z_{1}}|+|W^{u}|+|W^{v_{root}}|-2\leq k. \tag{8}
$$
Hence $|W^{z_{1}}|+|W^{u}|\leq k$ . Recall that $|W^{z_{1}}|\geq\lceil k/2\rceil$ . If $|W^{u}|\geq\lceil k/2\rceil$ , then $k$ is even and $|W^{z_{1}}|=|W^{u}|=k/2$ . By Claim 8, we have that $n_{1}(u)+n_{3}(u)-|R(u)|\leq\lfloor|W^{u}|/2\rfloor$ . Therefore, Ineq. (4) holds. Now, we assume that $2\leq|W^{u}|\leq\lceil k/2\rceil-1$ . Since $n_{4}(z_{1})\leq 1$ and $k-1\geq|W^{z_{1}}|\geq\lceil k/2\rceil$ , by Claim 5, there are at most $|W^{z_{1}}|-\left\lceil k/2\right\rceil+2$ internally disjoint $U^{s}$ -Steiner trees $T_{i}^{s}$ such that $d_{T_{i}^{s}}(u)=2$ (note that $u\in G_{s}$ ). Hence, $n_{4}(u)\leq|W^{z_{1}}|-\left\lceil k/2\right\rceil+2$ . Thus, by Eq. (5),
$$
n_{1}(u)+n_{3}(u)-|R(u)|=n_{4}(u)+|W^{u}|-\left\lceil k/2\right\rceil\leq|W^{u
}|+|W^{z_{1}}|-2\left\lceil k/2\right\rceil+2.
$$
If $|W^{u}|+|W^{z_{1}}|\leq k-1$ or $k$ is odd, then $n_{1}(u)+n_{3}(u)-|R(u)|\leq 1\leq\lfloor|W^{u}|/2\rfloor$ , Ineq. (4) holds. Thus, assume that $|W^{u}|+|W^{z_{1}}|=k$ (recall that $|W^{u}|+|W^{z_{1}}|\leq k$ ) and $k$ is even below ( $k\geq 4$ ). Since $|W^{z_{1}}|\geq\lceil k/2\rceil$ , it follows that $|W^{u}|\leq k/2$ . Suppose that $v_{root}\overrightarrow{T}z_{p}=v_{root}w_{1}w_{2}\ldots w_{q}v^{*}z_{p}$ . Since $|W^{v_{root}}|\geq 2$ , it follows from Ineq. (8) and Fact 1 that $|W^{v_{root}}|=2\leq k/2$ and $|W^{w_{1}}|=\ldots=|W^{w_{q}}|=|W^{v^{*}}|=1$ . According to Claim 7, we have that $n_{4}(u)=0$ . Hence,
| | $\displaystyle n_{1}(u)+n_{3}(u)-|R(u)|$ | $\displaystyle=n_{4}(u)+|W^{u}|-\left\lceil k/2\right\rceil\leq 0\leq\lfloor|W^ {u}|/2\rfloor,$ | |
| --- | --- | --- | --- |
the Ineq. (4) holds.
**Subcase 1.3**
*$p\geq 2$ .*
Then $u\succeq z_{3}$ . Recall that $|W^{z_{i}}|\geq 2$ for each $i\in[p]$ . Since
$$
\displaystyle|W^{v_{root}}|+|W^{z_{1}}|+|W^{z_{2}}|+|W^{z_{3}}|-3\leq k \tag{9}
$$
and $|W^{z_{1}}|\geq\lceil k/2\rceil$ , it follows that $|W^{z_{2}}|,|W^{z_{3}}|\leq\lfloor k/2\rfloor-1$ and $|W^{z_{1}}|+|W^{z_{2}}|\leq k-1$ . On the other hand, since $|W^{z_{3}}|\geq 2$ , it follows that $k\geq 6$ . Without loss of generality, suppose $z_{2}\in U^{\iota}$ . Recall Ineqs. (6) and (7), and combine with Claim 5, there are at most $|W^{z_{1}}|-\left\lceil\frac{k}{2}\right\rceil+2$ internally disjoint $U^{\iota}$ -Steiner trees $T_{i}^{\iota}$ such that $d_{T_{i}^{\iota}}(z_{2})=2$ . Hence, $n_{4}(z_{2})\leq|W^{z_{1}}|-\left\lceil\frac{k}{2}\right\rceil+2$ . By Claim 3, $n_{4}(z_{3})\leq\max\{1,n_{4}(z_{2})+|W^{z_{2}}|-\lceil k/2\rceil+1\}$ . However, by the maximality of $v^{*}$ , we have that $n_{4}(z_{3})\geq 2$ . Hence,
$$
\displaystyle n_{4}(z_{3})\leq n_{4}(z_{2})+|W^{z_{2}}|-\lceil k/2\rceil+1\leq
|W^{z_{1}}|+|W^{z_{2}}|-2\lceil k/2\rceil+3. \tag{10}
$$
Since $|W^{z_{1}}|+|W^{z_{2}}|\leq k-1$ , it follows that if $k$ is odd or $|W^{z_{1}}|+|W^{z_{2}}|\leq k-2$ , then $n_{4}(z_{3})\leq 1$ , a contradiction. Hence, $k$ is even and $|W^{z_{1}}|+|W^{z_{2}}|=k-1$ . This implies that $|W^{z_{3}}|=n_{4}(z_{3})=2$ by Ineqs. (9) and (10). Since $k\geq 6$ , it follows that $n_{4}(z_{3})+|W^{z_{3}}|-\lceil k/2\rceil\leq 1\leq\lfloor|W^{z_{3}}|/2\rfloor$ . Recall that $u\succeq z_{3}$ . If $v_{3}=u$ , then
$$
n_{1}(u)+n_{3}(u)-|R(u)|=n_{4}(u)+|W^{u}|-\lceil k/2\rceil\leq\lfloor|W^{u}|/2\rfloor,
$$
Ineq. (4) holds. If $u\succ z_{3}$ , then since
| | $\displaystyle|W^{v_{root}}|+|W^{z_{1}}|+|W^{z_{2}}|+|W^{z_{3}}|+|W^{u}|-4\leq k,$ | |
| --- | --- | --- |
we have that
| | $\displaystyle|W^{u}|\leq k+4-(|W^{v_{root}}|+|W^{z_{1}}|+|W^{z_{2}}|+|W^{z_{3} }|)=1,$ | |
| --- | --- | --- |
a contradiction.
**Case 2**
*$v^{*}$ does not exist.*
Let $\overrightarrow{P}=v_{root}\overrightarrow{T}u=v_{root}z_{1}\ldots,z_{t}u$ . Since $v^{*}$ does not exist, for each vertex $z\in V(\overrightarrow{P})$ , either $2\leq|W^{z}|\leq\lceil k/2\rceil-1$ and $n_{4}(z)\geq 2$ , or $|W^{z}|\geq\lceil k/2\rceil$ . Since $n_{4}(v_{root})=0$ , it follows that $|W^{v_{root}}|\geq\lceil k/2\rceil$ . By Claim 1, $n_{4}(z_{1})\leq|W^{v_{root}}|-\lceil k/2\rceil$ .
**Subcase 2.1**
*$|W^{z_{1}}|\geq\lceil k/2\rceil$ .*
Since $|W^{v_{root}}|,|W^{z_{1}}|\geq\lceil k/2\rceil$ , we have that $t\leq 1$ . Otherwise,
$$
\sum_{z\in V(\overrightarrow{P})}|W^{z}|-(|\overrightarrow{P}|-1)>k,
$$
which contradicts Fact 1. Moreover, if $t=1$ , then $k$ is even, $|W^{v_{root}}|=|W^{z_{1}}|=k/2$ and $|W^{u}|=2$ . Suppose that $t=0$ . Then $u=z_{1}$ , and hence $n_{4}(u)\leq|W^{v_{root}}|-\lceil k/2\rceil$ . Thus
| | $\displaystyle n_{1}(u)+n_{3}(u)-|R(u)|$ | $\displaystyle=n_{4}(u)+|W^{u}|-\lceil k/2\rceil$ | |
| --- | --- | --- | --- |
Ineq. (4) holds. Suppose $t=1$ . Then $\overrightarrow{P}=v_{root}z_{1}u$ . Hence, $k$ is even ( $k\geq 4$ ), $|W^{v_{root}}|=|W^{z_{1}}|=k/2$ and $|W^{u}|=2$ . By the first statement of Claim 8 (here, we regard $v_{root}$ and $z_{1}$ as the vertices $b$ and $a$ in Claim 8, respectively, and then $u$ can be regarded as the vertex $x$ in Claim 8), we have that $n_{4}(u)\leq 1$ . Hence,
$$
n_{1}(u)+n_{3}(u)-|R(u)|=n_{4}(u)+|W^{u}|-\lceil k/2\rceil\leq 1\leq\lfloor|W^
{u}|/2\rfloor,
$$
Ineq. (4) holds.
**Subcase 2.2**
*$2\leq|W^{z_{1}}|\leq\lceil k/2\rceil-1$ .*
Recall that $n_{4}(z_{1})=|W^{v_{root}}|-\lceil k/2\rceil$ . Since $n_{4}(z_{1})\geq 2$ , $|W^{v_{root}}|\geq\lceil k/2\rceil+2$ . If $|W^{v_{root}}|+|W^{z_{1}}|=k+1$ , then $v=v_{root}$ , $u=z_{1}$ and
$$
n_{1}(u)+n_{3}(u)-|R(u)|=n_{4}(u)+|W^{u}|-\lceil k/2\rceil\leq 1\leq\lfloor|W^
{u}|/2\rfloor.
$$
Thus, suppose $|W^{v_{root}}|+|W^{z_{1}}|\leq k$ . Then
$$
n_{1}(z_{1})+n_{3}(z_{1})-|R(z_{1})|=n_{4}(z_{1})+|W^{z_{1}}|-\lceil k/2\rceil
\leq 0.
$$
If $z_{1}=u$ , then $n_{1}(u)+n_{3}(u)-|R(u)|\leq 0,$ Ineq. (4) holds. Now, assume that $z_{1}\neq u$ . Then $z_{2}$ exists. By Claim 3, $n_{4}(z_{2})\leq 1$ . Since $z_{2}$ is not a candidate of $v^{*}$ , it follows that $|W^{z_{2}}|\geq\lceil k/2\rceil$ . Since $|W^{v_{root}}|\geq\lceil k/2\rceil+2,|W^{z_{2}}|\geq\lceil k/2\rceil$ and $|W^{z_{1}}|\geq 2$ , it follows that $|W^{v_{root}}|+|W^{z_{1}}|+|W^{z_{2}}|-2\geq k+2$ , which contradicts Fact 1. ∎
With the conclusion of Lemma 2.1, the proof of Theorem 2.1 is completed.
### 2.3 Proof of Theorem 1.5
We first consider $3\leq k\leq\ell$ . The lower bound $\ell-\lceil k/2\rceil\leq\kappa_{k}(S(n,\ell))\leq\lambda_{k}(S(n,\ell))$ can be obtained from Theorem 2.1 directly. For the upper bounds of $\kappa_{k}(S(n,\ell))$ and $\lambda_{k}(S(n,\ell))$ , consider the graph $G_{n-1}$ . Let $V(G_{n-1})=\{u_{1},\ldots,u_{\ell}\}$ , $U=\{x_{1},\ldots,x_{k}\}$ and $x_{i}\in A_{n-1,u_{i}}$ , where $k\leq\ell$ . Suppose there are $p$ edge-disjoint $U$ -Steiner trees $T^{\prime}_{1},\ldots,T^{\prime}_{p}$ of $G=S(n,\ell)$ and $\mathcal{P}=\{A_{n-1,u}:u\in V(G_{n-1})\}$ . Let $T^{*}_{i}=T^{\prime}_{i}/\mathcal{P}$ for $i\in[p]$ . Then $T^{*}_{1},\ldots,T^{*}_{p}$ are edge-disjoint connected graphs of $G_{n-1}$ containing $\{u_{1},\ldots,u_{k}\}$ . Thus, $p\leq\lambda_{k}(G_{n-1})=\lambda_{k}(K_{\ell})=\ell-\lceil k/2\rceil$ . Therefore, $\kappa_{k}(S(n,\ell))\leq\lambda_{k}(S(n,\ell))\leq\ell-\lceil k/2\rceil$ , the upper bound follows.
Now consider the case $\ell+1\leq k\leq{\ell}^{n}$ . By Theorem 1.2, $\lambda_{k}(S(n,\ell))\geq\lfloor\ell/2\rfloor$ . Since $\lambda_{k}(S(n,\ell))\leq\lambda_{\ell}(S(n,\ell))=\lfloor\ell/2\rfloor$ , it follows that $\kappa_{k}(S(n,\ell))\leq\lambda_{k}(S(n,\ell))\leq\lfloor\ell/2\rfloor$ . Therefore, $\lambda_{k}(S(n,\ell))=\lfloor\ell/2\rfloor$ and $\kappa_{k}(S(n,\ell))\leq\lfloor\ell/2\rfloor$ . The proof is completed.
## 3 Some network properties
Generalized connectivity is a graph parameter to measure the stability of networks. In the following part, we will give the following other properties of Sierpiński graphs.
The Sierpiński graph(networks) is obtained after $t$ iteration as $S(t,\ell)$ that has $N_{t}$ nodes and $E_{t}$ edges, where $t=0,1,2,\cdots,T-1$ , and $T$ is the total number of iterations, and our generation process can be illustrated as follows.
<details>
<summary>x3.png Details</summary>
### Visual Description
## Line Graph: Exponential Functions Comparison
### Overview
The image is a line graph comparing two exponential functions: $ 3^t $ (blue line) and $ \frac{1}{2}(3^{t+1} - 3) $ (orange line). The x-axis represents the variable $ t $, ranging from 0 to 14, while the y-axis represents the function values, scaled up to $ 2.5 \times 10^6 $. The graph highlights the growth patterns of both functions, with the orange line consistently outpacing the blue line after $ t = 1 $.
### Components/Axes
- **X-axis (t)**: Labeled as $ t $, with tick marks at intervals of 2 (0, 2, 4, ..., 14).
- **Y-axis (Value)**: Labeled as "Value" (no explicit unit), with tick marks at $ 5 \times 10^5 $, $ 1 \times 10^6 $, $ 1.5 \times 10^6 $, $ 2 \times 10^6 $, and $ 2.5 \times 10^6 $.
- **Legend**: Located on the right side of the graph.
- **Blue line**: Labeled $ 3^t $.
- **Orange line**: Labeled $ \frac{1}{2}(3^{t+1} - 3) $.
### Detailed Analysis
- **Blue Line ($ 3^t $)**:
- Starts at $ 1 $ when $ t = 0 $.
- Grows exponentially, with values doubling approximately every $ t \approx 1.58 $ (since $ 3^t $ doubles every $ \log_3(2) \approx 0.63 $, but the graph shows a slower visual doubling due to scaling).
- At $ t = 10 $, $ 3^{10} = 59,049 $, which is near the lower end of the y-axis.
- At $ t = 14 $, $ 3^{14} = 4,782,969 $, exceeding the y-axis maximum of $ 2.5 \times 10^6 $.
- **Orange Line ($ \frac{1}{2}(3^{t+1} - 3) $)**:
- Simplifies to $ \frac{3^{t+1} - 3}{2} = \frac{3 \cdot 3^t - 3}{2} = \frac{3}{2}(3^t - 1) $.
- Starts at $ 0 $ when $ t = 0 $ (since $ \frac{3^{1} - 3}{2} = 0 $).
- At $ t = 1 $, both lines intersect at $ 3 $.
- For $ t > 1 $, the orange line grows faster than the blue line. For example:
- At $ t = 2 $: Blue = $ 9 $, Orange = $ \frac{27 - 3}{2} = 12 $.
- At $ t = 10 $: Blue = $ 59,049 $, Orange = $ \frac{3^{11} - 3}{2} = \frac{177,147 - 3}{2} = 88,572 $.
- At $ t = 14 $: Orange = $ \frac{3^{15} - 3}{2} = \frac{14,348,907 - 3}{2} \approx 7,174,452 $, far exceeding the y-axis range.
### Key Observations
1. **Intersection at $ t = 1 $**: Both functions equal $ 3 $ at this point.
2. **Growth Rate Difference**: The orange line grows 1.5 times faster than the blue line for $ t > 1 $, as $ \frac{3}{2}(3^t - 1) $ scales the blue line by 1.5 and subtracts a constant.
3. **Y-axis Limitation**: The graph truncates values beyond $ 2.5 \times 10^6 $, obscuring the full extent of the orange line’s growth.
### Interpretation
The graph demonstrates how exponential functions with different scaling factors behave. The orange line, $ \frac{1}{2}(3^{t+1} - 3) $, is a transformed version of $ 3^t $, scaled by 1.5 and shifted downward by 1.5. This transformation ensures the orange line starts at 0 (vs. 1 for the blue line) and grows faster after $ t = 1 $. The intersection at $ t = 1 $ suggests a critical point where the two models yield equal results, potentially significant in contexts like population growth, financial models, or algorithmic complexity. The y-axis truncation highlights the rapid divergence between the two functions, emphasizing the importance of scaling in exponential growth scenarios.
</details>
Figure 7: The Function of $E_{t}=3^{t}$ and $N_{t}=\frac{3^{t+1}-3}{2}$
Step 1: Initialization. Set $t=0$ , $G_{1}$ is a complete graph of order $\ell$ , and thus $N_{1}=\ell$ and $E_{1}=\binom{\ell}{2}$ . Set $G_{1}=S(1,\ell)$ .
Step 2: Generation of $G_{t+1}$ from $G_{t}$ . Let $S^{1}(t,\ell),S^{2}(t,\ell),\ldots,S^{\ell}(t,\ell)$ be all Sierpiński graphs added at Step $t$ , where $S^{i}(t,\ell)\cong S(t,\ell)$ ( $1\leq i\leq\ell$ ). At Step $t+1$ , we add one edge (bridge edge) between $S^{i}(t,\ell)$ and $S^{j}(t,\ell)$ , $i\neq j$ , namely the edge between vertices $\langle ij\cdots j\rangle$ and $\langle ji\cdots i\rangle$ .
For Sierpiński graphs $G_{t}=S(t,\ell)$ , its order and size are $N_{t}=\ell^{t}$ and $E_{t}=\frac{\ell^{t+1}-\ell}{2}$ , respectively; see Table 2 and Figure 7 (for $\ell=3$ ).
Table 2: The size and order of $G_{t}$ for $\ell=3$
| $t$ | 1 | 2 | 3 | 4 | 5 | 6 | t |
| --- | --- | --- | --- | --- | --- | --- | --- |
| $N_{t}$ | 3 | 9 | 27 | 81 | 242 | 729 | $\ell^{t}$ |
| $E_{t}$ | 3 | 12 | 39 | 120 | 363 | 1092 | $\frac{\ell^{t+1}-\ell}{2}$ |
The degree distribution for $t$ times are
$$
\left(\underbrace{\ell-1,\ell-1,\cdots,\ell-1}_{\ell},\underbrace{\ell,\ell,
\ell,\cdots,\ell}_{\ell^{t}-\ell}\right). \tag{11}
$$
From Equation 11, the instantaneous degree distribution is $P(\ell-1,t)=1/\ell^{t-1}$ for $t=2,\cdots,T$ and $P(\ell,t)=(\ell^{t}-\ell)/\ell^{t}$ for $t=2,\cdots,T$ . Note that the density of Sierpiński graphs is $\rho=E_{t}/\binom{N_{t}}{2}\rightarrow 0$ for $t\rightarrow+\infty$ . For large enough $\ell$ and any $1\leq k\leq\ell$ , we have $|\{v\in V(G)|d_{S(t,\ell)}(v)|\geq k\}\approx|V(S(t,\ell))|$ .
**Theorem 3.1**
*[11] If $n\in\mathbb{N}$ and $G$ is a graph, then $\kappa\left(S(n,G)\right)=\kappa(G)$ and $\lambda\left(S(n,G)\right)=\lambda(G)$ .*
From Theorem 3.1, we have $\kappa\left(S(n,\ell)\right)=\kappa(K_{\ell})=\ell-1$ and $\lambda\left(S(n,\ell)\right)=\lambda(K_{\ell})=\ell-1$ . Note that $\lambda_{k}(S(n,\ell))=\ell-\left\lceil k/2\right\rceil$ and $\kappa_{k}(S(n,\ell))=\ell-\left\lceil k/2\right\rceil$ ; see Figure 8.
<details>
<summary>x4.png Details</summary>
### Visual Description
## 3D Plot: Relationships Between I, I - [k/2], and k
### Overview
The image depicts a 3D plot with three intersecting planes representing mathematical relationships between variables **I**, **k**, and their combination **I - [k/2]**. The axes are labeled **x**, **y**, and **z**, each ranging from **-10 to 30**. The legend in the top-right corner maps colors to equations: **blue** for **I - [k/2]**, **orange** for **I**, and **green** for **k**. The planes exhibit distinct spatial trends, with the blue plane sloping downward, the orange plane sloping upward, and the green plane remaining flat.
---
### Components/Axes
- **Axes**:
- **x-axis**: Labeled "x", ranging from **-10 to 30**.
- **y-axis**: Labeled "y", ranging from **-10 to 30**.
- **z-axis**: Labeled "z", ranging from **-10 to 30**.
- **Legend**: Positioned in the **top-right corner**, with:
- **Blue**: Equation **I - [k/2]** (diagonal plane).
- **Orange**: Equation **I** (sloped plane).
- **Green**: Equation **k** (flat plane).
---
### Detailed Analysis
1. **Blue Plane (I - [k/2])**:
- **Equation**: **z = I - [k/2]**.
- **Trend**: Diagonal plane sloping **downward** from the origin (0,0,0) toward negative **z** values as **x** and **y** increase.
- **Key Points**:
- At **x = 0, y = 0**: **z ≈ 0** (intersects origin).
- At **x = 30, y = 30**: **z ≈ -15** (approximate extrapolation based on slope).
- **Spatial Grounding**: Dominates the lower-left quadrant of the plot.
2. **Orange Plane (I)**:
- **Equation**: **z = I**.
- **Trend**: Sloped plane rising **upward** from the origin, with **z** increasing linearly with **x** and **y**.
- **Key Points**:
- At **x = 0, y = 0**: **z ≈ 0** (intersects origin).
- At **x = 30, y = 30**: **z ≈ 30** (approximate extrapolation).
- **Spatial Grounding**: Dominates the upper-right quadrant.
3. **Green Plane (k)**:
- **Equation**: **z = k** (constant).
- **Trend**: Flat plane at **z ≈ 30**, parallel to the **x-y** plane.
- **Key Points**:
- Uniform **z = 30** across all **x** and **y** values.
- **Spatial Grounding**: Covers the top surface of the plot.
---
### Key Observations
- **Intersection**: All three planes intersect at the origin **(0,0,0)**.
- **Color Consistency**: Legend colors match the planes:
- Blue (**I - [k/2]**) aligns with the downward-sloping plane.
- Orange (**I**) aligns with the upward-sloping plane.
- Green (**k**) aligns with the flat plane at **z = 30**.
- **Anomalies**: No outliers or discontinuities observed; all planes are continuous.
---
### Interpretation
The plot illustrates how **I**, **k**, and their adjusted combination (**I - [k/2]**) interact spatially. The **blue plane** (I - [k/2]) suggests a **diminishing relationship** between **I** and **k**, as increasing **k** reduces the value of **I - [k/2]**. The **orange plane** (I) represents a direct proportionality between **I** and spatial coordinates **x/y**, while the **green plane** (k) indicates **k** is a constant parameter independent of **x** and **y**. This could model a system where **I** and **k** are interdependent variables, with **I - [k/2]** acting as a corrected or normalized metric. The flat **k** plane implies **k** is fixed, possibly representing a boundary condition or threshold in the system.
</details>
Figure 8: The generalized (edge-)connectivity of $S(n,\ell)$
The number of spanning tree of $G$ denoted by $\tau(G)$ . Let
$$
\rho(G)=\lim_{V(G)\longrightarrow\infty}\frac{\ln|\tau(G)|}{\left|V(G)\right|}, \tag{12}
$$
where $\rho(G)$ is called the entropy of spanning trees or the asymptotic complexity [2, 7].
As an application of generalized (edge-)connectivity, similarly to the Equation 12, it can describe the fault tolerance of a graph or network, a common metric is called the entropy of spanning trees.
we give the entropy of the $k$ -Steiner tree of a graph $G$ can be defined as
$$
\rho_{k}(G)=\lim_{|V(G)|\longrightarrow\infty}\frac{\ln|\kappa_{k}(G)|}{\left|
V(G)\right|}
$$
The entropy of the $3,6,9$ -Steiner tree of Sierpiński graph $S(8,\ell)$ can be seen in Figure 9.
<details>
<summary>x5.png Details</summary>
### Visual Description
## Line Graph: Comparison of Logarithmic Functions Divided by 8th Root
### Overview
The image is a line graph comparing three logarithmic functions divided by the eighth root of a variable. The x-axis ranges from 3 to 9, and the y-axis spans from -2×10⁻⁶ to 4×10⁻⁶. Three distinct lines (blue, orange, green) represent different logarithmic expressions, with a legend on the right correlating colors to their respective functions.
### Components/Axes
- **X-axis**: Labeled with integer values from 3 to 9 (no explicit title).
- **Y-axis**: Labeled with scientific notation values (-2×10⁻⁶ to 4×10⁻⁶), no explicit title.
- **Legend**: Positioned on the right, with three entries:
- **Blue**: `log(-3/2)/8`
- **Orange**: `log(-6/2)/8`
- **Green**: `log(-9/2)/8`
### Detailed Analysis
1. **Blue Line (`log(-3/2)/8`)**:
- **Behavior**: Vertical asymptote at x=3 (infinite value), then sharply decreases.
- **Trend**: Dominates the left side of the graph, dropping rapidly toward the x-axis as x increases.
- **Key Points**:
- At x=3: Undefined (asymptote).
- At x=4: ~4×10⁻⁶.
- At x=5: ~2×10⁻⁶.
- At x=6: ~1×10⁻⁶.
2. **Orange Line (`log(-6/2)/8`)**:
- **Behavior**: Peaks at x=4.5 (~2.5×10⁻⁶), then declines smoothly.
- **Trend**: Crosses the blue line near x=5, converging with it as x increases.
- **Key Points**:
- At x=4: ~2×10⁻⁶.
- At x=5: ~1.5×10⁻⁶.
- At x=6: ~1×10⁻⁶.
3. **Green Line (`log(-9/2)/8`)**:
- **Behavior**: Starts negative (~-2×10⁻⁶ at x=3), crosses zero near x=6, then becomes positive.
- **Trend**: Gradually rises after x=6, intersecting the orange line near x=7.
- **Key Points**:
- At x=3: ~-2×10⁻⁶.
- At x=6: ~0.
- At x=7: ~1×10⁻⁶.
### Key Observations
- **Asymptotic Behavior**: The blue line exhibits a vertical asymptote at x=3, suggesting a division-by-zero scenario in its denominator.
- **Convergence**: All lines converge toward the x-axis as x increases, indicating diminishing magnitudes.
- **Zero Crossing**: The green line crosses the x-axis near x=6, transitioning from negative to positive values.
- **Peak and Intersection**: The orange line peaks at x=4.5 and intersects the blue line near x=5.
### Interpretation
The graph illustrates how logarithmic functions scaled by the eighth root of a variable behave across a domain. The vertical asymptote at x=3 for the blue line implies a critical point where the denominator approaches zero, causing the function to diverge. The orange and green lines demonstrate how varying the numerator (e.g., -3/2 vs. -9/2) affects the function's growth rate and zero-crossing behavior. The convergence of all lines toward zero as x increases suggests that the eighth root term dominates the logarithmic growth at larger x-values.
**Notable Anomaly**: The logarithmic arguments (-3/2, -6/2, -9/2) are negative, which would make the functions undefined in real-number systems. This could indicate a typo (e.g., absolute values) or a complex-number context not reflected in the graph. The y-axis values (extremely small magnitudes) further complicate interpretation, as typical logarithmic functions would not produce such results without additional scaling or constraints.
</details>
Figure 9: Entropy of $S(n,\ell)$ for $n=8$ , $k=3,6,9$ and $\ell\rightarrow+\infty$
The definition of clustering coefficient can be found in [3]. Let $N_{v}(t)$ be the number of edges in $G_{t}$ among neighbors of $v$ , which is the number of triangles connected to the vertex $v$ . The clustering coefficient of a graph is based on a local clustering coefficient for each vertex
$$
C[v]=\frac{N_{v}(t)}{d_{G}(v)(d_{G}(v)-1)/2},
$$
If the degree of node $v$ is $0 0$ or $1$ , then we can set $C[v]=0$ . By definition, we have $0\leq C[v]\leq 1$ for $v\in V(G)$ .
The clustering coefficient for the whole graph $G$ is the average of the local values $C(v)$
$$
C(G)=\frac{1}{|V(G)|}\left(\sum_{v\in V(G)}C[v]\right).
$$
The clustering coefficient of a graph is closely related to the transitivity of a graph, as both measure the relative frequency of triangles [22, 24].
**Proposition 3.1**
*The clustering coefficient of generalized Sierpiński graph $S(n,\ell)$ is
$$
C(S(n,\ell))=\frac{\ell^{-n}\left(2\ell-2\ell^{n}+\ell^{n+1}\right)}{\ell}.
$$*
* Proof*
For any $v\in V(S(n,\ell))$ , if $v$ is a extremal vertex, then $d_{S(n,\ell)}(v)=\ell-1$ and $G[\{N(v)\}]\cong K_{\ell-1}$ , and hence
$$
C[v]=\frac{N_{v}(t)}{d_{G}(v)(d_{G}(v)-1)/2}={\binom{\ell-1}{2}}/{\binom{\ell-
1}{2}}=1.
$$
If $v$ is not a extremal vertex, then $d_{S(n,\ell)}(v)=\ell$ and $G[\{N(v)\}]\cong K_{\ell}+e$ , where $K_{\ell}+e$ is graph obtained from a complete graph $K_{\ell}$ by adding a pendent edge. Hence, we have $C[v]={\binom{\ell-1}{2}}/{\binom{\ell}{2}}=\frac{\ell-2}{\ell}$ . Since there exists $\ell$ extremal vertices in Sierpiński graph $S(n,\ell)$ , it follows that
$$
C(S(n,\ell))=\frac{1}{|V(G)|}\left(\sum_{v\in V(G)}C[v]\right)=\frac{1}{\ell^{
n}}\left({\ell\times 1+(\ell^{n}-\ell)\frac{\ell-2}{\ell}}\right)=\frac{\ell^{
-n}\left(2\ell-2\ell^{n}+\ell^{n+1}\right)}{\ell}
$$
∎
<details>
<summary>x6.png Details</summary>
### Visual Description
## Line Graph: C(S(3,I)) vs I
### Overview
The image depicts a line graph illustrating the relationship between a variable labeled "I" (x-axis) and a function "C(S(3,I))" (y-axis). The graph shows a decreasing trend in "C(S(3,I))" as "I" increases, with a sharp vertical segment at the origin.
### Components/Axes
- **X-axis**: Labeled "I", scaled from 0 to 100 in increments of 20.
- **Y-axis**: Labeled "C(S(3,I))", scaled from 0.8 to 1.1 in increments of 0.1.
- **Legend**: Located at the top-right corner, indicating the line represents "C(S(3,I))" in blue.
- **Line**: A single blue curve starting at (0, 1.1), curving downward, and approaching a horizontal asymptote near y = 0.95 as x increases.
### Detailed Analysis
- **Vertical Segment**: At x = 0, the line extends vertically from y = 0.8 to y = 1.1.
- **Curve Behavior**: The curve decreases sharply initially, then flattens as x increases. Key approximate points:
- At x = 0: y = 1.1 (peak).
- At x = 20: y ≈ 0.92.
- At x = 40: y ≈ 0.94.
- At x = 60: y ≈ 0.95.
- At x = 80: y ≈ 0.96.
- At x = 100: y ≈ 0.97.
### Key Observations
- The function "C(S(3,I))" exhibits a **saturation effect**, where its value decreases rapidly at low "I" and stabilizes near 0.95–0.97 for higher "I".
- The vertical segment at x = 0 suggests a discontinuity or undefined behavior at the origin.
### Interpretation
The graph likely models a system where "C(S(3,I))" represents a capacity, efficiency, or probability metric that diminishes as "I" (possibly an input or intensity parameter) increases. The saturation behavior implies diminishing returns or a limiting factor in the system. The vertical segment at x = 0 may indicate an initial condition or boundary condition where the function is undefined or capped. The trend aligns with models where variables approach an asymptotic limit under increasing inputs.
</details>
Figure 10: The Function of $C_{S(3,l)}$
<details>
<summary>x7.png Details</summary>
### Visual Description
## Line Graph: D(S(n,l)) vs l
### Overview
The image depicts a line graph illustrating the relationship between the variable `l` (x-axis) and the function `D(S(n,l))` (y-axis). The graph shows a gradual increase in `D(S(n,l))` for `l` values from 2 to 10, followed by a sharp exponential rise for `l` values from 10 to 14. The y-axis scales from 0 to 30,000, while the x-axis ranges from 2 to 16.
---
### Components/Axes
- **X-axis (Horizontal)**: Labeled `l`, with discrete markers at intervals of 2 (2, 4, 6, ..., 16).
- **Y-axis (Vertical)**: Labeled `D(S(n,l))`, with increments of 5,000 (0, 5,000, 10,000, ..., 30,000).
- **Legend**: Located in the top-right corner, labeled `D(S(n,l))` with a blue line.
- **Line**: A single blue line representing `D(S(n,l))`, plotted across the x-axis range.
---
### Detailed Analysis
1. **Initial Trend (l = 2 to 10)**:
- The line remains nearly flat, with `D(S(n,l))` values close to 0.
- At `l = 10`, the value is approximately **1,000** (estimated from the y-axis scale).
2. **Exponential Rise (l = 10 to 14)**:
- The line steeply ascends, reaching **25,000** at `l = 14`.
- The slope becomes increasingly nonlinear, suggesting a power-law or exponential growth pattern.
3. **Post-14 Behavior**:
- The graph does not extend beyond `l = 14`, but the y-axis suggests potential values up to 30,000.
---
### Key Observations
- **Threshold Effect**: A critical transition occurs at `l = 10`, where `D(S(n,l))` begins to increase rapidly.
- **Data Sparsity**: Only two distinct data points are clearly marked: `(l=10, D≈1,000)` and `(l=14, D=25,000)`.
- **Uncertainty**: Values for `l < 10` are approximated due to the line’s proximity to the baseline.
---
### Interpretation
The graph suggests that `D(S(n,l))` is relatively stable for `l ≤ 10`, possibly indicating a system in equilibrium or a baseline state. Beyond `l = 10`, the function exhibits a dramatic increase, implying a phase transition, critical threshold, or nonlinear dependency on `l`. This could represent phenomena such as:
- **System Instability**: A sudden breakdown or amplification in a physical or computational system.
- **Combinatorial Explosion**: Exponential growth in complexity (e.g., algorithmic or network behavior).
- **Phase Transition**: A material or thermodynamic property changing abruptly at a critical point.
The absence of data beyond `l = 14` leaves the long-term behavior of `D(S(n,l))` ambiguous, but the sharp rise hints at a potential saturation or failure mode. Further data points would clarify whether the growth continues exponentially or plateaus.
</details>
Figure 11: The diameter of $S(n,\ell)$
**Theorem 3.2**
*[21] The diameter of $S(n,\ell)$ is $Diam(S(n,\ell))=2^{\ell}-1$ ;*
For network properties of Sierpiński graph $S(n,\ell)$ , the the diameter function can be seen in Figure 11 and its clustering coefficient is closely related to $1$ when $\ell\longrightarrow\infty$ ; see Figure 11, which implies that the Sierpiński graph $S(n,\ell)$ is a hight transitivity graph.
## 4 Acknowledgments
The authors would like to thank the anonymous referee for the careful reading of the manuscript and the numerous helpful suggestions and comments.
## References
- [1] J.A. Bondy, U.S.R. Murty, Graph Theory, GTM 244, Springer, 2008.
- [2] R. Burton, R. Pemantle, Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances, Ann. Prob. 21 (1993), 1329–1371.
- [3] B. Bollobás, O.M. Riordan, Mathematical results on scale-free random graphs, in: S. Bornholdt, H.G. Schuster (Eds.), Handbook of Graphs and Networks: From the Genome to the Internet, Wiley-VCH, Berlin, 2003, 1–34.
- [4] G. Chartrand, F. Okamoto, P. Zhang, Rainbow trees in graphs and generalized connectivity, Networks 55(4) (2010), 360–367.
- [5] B.L. Chen, K.C. Huange, On the Line $k$ -arboricity of $K_{n}$ and $K_{n,n}$ , Discrete Math. 254 (2002), 83–87.
- [6] X. Cheng, D. Du, Steiner trees in Industry, Kluwer Academic Publisher, Dordrecht, 2001.
- [7] M. Dehmer, F. Emmert-Streib, Z. Chen, X. Li, and Y. Shi, Mathematical foundations and applications of graph entropy, John Wiley $\&$ Sons. (2017).
- [8] D. Du, X. Hu, Steiner tree problems in computer communication networks, World Scientific, 2008.
- [9] C. Hao, W. Yang, The generalized connectivity of data center networks. Parall. Proce. Lett. 29 (2019), 1950007.
- [10] J. Gu, J. Fan, Q. Ye, L. Xi, Mean geodesic distance of the level- $n$ Sierpiński gasket, J. Math. Anal. Appl. 508(1) (2022), 125853.
- [11] S. Klavžar, U. Milutinović, Graphs $S(n,k)$ and a variant of tower of hanoi problem, Czechoslovak Math. J. 122(47) (1997), 95–104.
- [12] H. Li, X. Li, and Y. Sun, The generalied $3$ -connectivity of Cartesian product graphs, Discrete Math. Theor. Comput. Sci. 14(1) (2012), 43–54.
- [13] H. Li, Y. Ma, W. Yang, Y. Wang, The generalized $3$ -connectivity of graph products, Appl. Math. Comput. 295 (2017), 77–83.
- [14] S. Li, J. Tu, C. Yu, The generalized $3$ -connectivity of star graphs and bubble-sort graphs, Appl. Math. Comput. 274(1) (2016), 41–46.
- [15] X. Li, Y. Mao, Generalized Connectivity of Graphs, Springer Briefs in Mathematics, Springer, Switzerland, 2016.
- [16] X. Li, Y. Mao, Y. Sun, On the generalized (edge-)connectivity of graphs, Australasian J. Combin. 58(2) (2014), 304–319.
- [17] Y. Li, Y. Mao, Z. Wang, Z. Wei, Generalized connectivity of some total graphs, Czechoslovak Math. J. 71 (2021), 623–640.
- [18] C. Li, S. Lin, S. Li, The $4$ -set tree connectivity of $(n,k)$ -star networks, Theor. Comput. Sci. 844 (2020), 81–86.
- [19] S. Lin, Q. Zhang, The generalized $4$ -connectivity of hypercubes. Discrete Appl. Math. 220 (2017), 60–67.
- [20] H. Li, X. Li, Y. Sun, The generalized $3$ -connectivity of Cartesian product graphs, Discrete Math. Theor. Comput. Sci. 14 (2012), 43–54.
- [21] D. Parisse, On some metric properties of the the Sierpiński graphs $S(n,\ell)$ , Ars Comb. 90 (2009), 145–160.
- [22] J. Reichardt, Structure in complex networks. Springer Science $\&$ Business Media, 2008.
- [23] A. Sahu, A. Priyadarshi, On the box-counting dimension of graphs of harmonic functions on the Sierpiński gasket. J. Math. Analy. Appl. 487(2) (2020), 124036.
- [24] X. Wang, X. LI, G. R.Chen, Introduction to Network Science. Beijing: Higher Education Press, 2012.
- [25] B. Xue, L. Zuo, G. Li, The hamiltonicity and path $t$ -coloring of Sierpiński-like graphs, Discrete Appl. Math. 160 (2012), 1822–836.
- [26] B. Xue, L. Zuo, On the linear $(n-1)$ -arboricity of $K_{n}$ , Discrete Appl. Math. 158 (2010) 1546–1550.
- [27] S. Zhao, R. Hao, L. Wu, The generalized connectivity of $(n,k)$ -bubble-sort graphs. The Comput. J. 62 (2019), 1277–1283.
- [28] S. Zhao, R. Hao, J. Wu, The generalized $3$ -connectivity of some regular networks, J. Parall. Distr. Comput. 133 (2019), 18–29.
- [29] S. Zhao, R. Hao, The generalized connectivity of alternating group graphs and $(n,k)$ -star graphs, Discrete Appl. Math. 251 (2018), 310–321.
- [30] S. Zhao, R. Hao, E. Cheng, Two kinds of generalized connectivity of dual cubes, Discrete Appl. Math. 257 (2019), 306–316.
- [31] S. Zhao, R. Hao, The generalized connectivity of alternating group graphs and $(n,k)$ -star graphs, Discrete Appl. Math. 251 (2018), 310–321.
- [32] S. Zhao, R. Hao, The generalized $4$ -connectivity of exchanged hypercubes, Applied Mathe. Comput. 347 (2019), 342–353.
- [33] S. Zhao, R. Hao, J. Wu, The generalized $4$ -connectivity of hierarchical cubic networks, Discrete Appl. Mathe., 289 (2021), 194–206.
- [34] S. Zhao, J. Chang, Z. Li, The generalized $4$ -connectivity of pancake graphs, Discrete Appl. Math. 327 (2023), 77–86.
- [35] S. Zhao, R. Hao, The generalized $4$ -connectivity of exchanged hypercubes. Appl. Math. Comput. 347 (2019), 342–353.