Linear algorithms on Steiner domination of trees

TL;DR

This paper introduces a linear-time algorithm for finding the minimum Steiner dominating set in trees, advancing the efficiency of solving this graph domination problem.

Contribution

It presents the first linear algorithm specifically designed for computing the Steiner domination number in trees, improving computational efficiency.

Findings

The algorithm computes minimum Steiner dominating sets in linear time.

It demonstrates improved efficiency over previous methods.

The approach is applicable to various tree structures.

Abstract

A set of vertices $W$ in a connected graph $G$ is called a Steiner dominating set if $W$ is both Steiner and dominating set. The Steiner domination number $\gamma_{st}(G)$ is the minimum cardinality of a Steiner dominating set of $G$. A linear algorithm is proposed in this paper for finding a minimum Steiner dominating set for a tree $T$.

Full text

††journal: Discrete Applied Mathematics

1 Introduction

In this paper, we only consider finite, connected and undirected graph $G$. We refer to the books [1, 2] for notation and terminology on graph theory and theory of domination.

Let $G=(V(G),E(G))$ be a graph with the order of vertex set $|V(G)|$ and the order of edge set $|E(G)|$. The open neighborhood and the closed neighborhood of a vertex $v\in V$ are denoted by $N(v)=\{{u\in V(G):vu\in E(G)}\}$ and $N[v]=N(v)\cup\{v\}$, respectively. For a vertex set $S\in V(G)$, $N(S)=\bigcup\limits_{v\in S}{N(v)}$, and $N[S]=\bigcup\limits_{v\in S}{N[V]}$. The distance $d(u,v)$ between two vertices $u$ and $v$ of a connected graph $G$ is the length of shortest $u-v$ path in $G$. For a non-empty set $W$ of vertices in connected graph $G$, the Steiner distance $d(W)$ of $W$ is the minimum size of a connected subgraph of $G$ containing $W$. Obviously, each such subgraph is a tree and is called a Steiner tree or a Steiner $W$-tree. The set of all vertices of $G$ that lie on some Steiner $W$-tree is denoted by $S(W)$. If $S(W)=V(G)$ then $W$ is called Steiner set of $G$. The Steiner number $s(G)$ is the minimum cardinality of a Steiner set.

Chartrand and Zhang introduced the concept of Steiner number of a connected graph $G$ in [3]. Pelayo corrected main result in [4]. He proved that not all Steiner sets are geodetic sets and there are connected graphs whose Steiner number is strictly lower than their geodetic number. Hernando et al. [5] have studied the relationships between Steiner sets and geodetic sets and between Steiner sets and monophonic sets. Many results on Steiner distance were given in [6, 7].

A subset $S$ of V(G) is called dominating set if every vertex $v\in V$ is either a vertex of $S$ or is adjacent to a vertex of $S$. The domination number $\gamma(G)$ is the minimum cardinality of minimal dominating set of $G$. A systematic visit of each vertex of a tree is called a tree traversal. A set of vertices $W$ in a connected graph $G$ is called a Steiner dominating set if $W$ is both Steiner and dominating set. The Steiner domination number $\gamma_{st}(G)$ is the minimum cardinality of a Steiner dominating set of $G$.

The concept of Steiner domination was introduced in [8], and Vaidya etc. have obtained various results on Steiner domination numbers in [9, 10, 11].

The most algorithmic complexity of domination and related parameters of graphs are NP-complete or NP-hard problems. But there are many linear algorithms for domination and related parameters in trees, such as domination, total domination and secure domination in trees [12, 13, 14]. In this paper, we present a linear algorithm of Steiner domination in trees. It is similar to an algorithm due to Mitchell, Cockayne and Hedetniemi [15] for computing the domination number of an arbitrary tree.

2 Lemmas

A vertex of a graph $G$ is called a leaf or end-vertex if it is adjacent to only one vertex in $G$. A vertex $v$ is an extreme vertex if the subgraph induced by its neighbors is complete. Thus, every end-vertex is an extreme vertex.

** Lemma 2.1**

[3]** Each extreme vertex of a graph G belongs to every Steiner set of G. In particular, each end-vertex of G belongs to every Steiner set of G.

The following corollary is an immediate consequence of Lemma 2.1.

** Corollary 2.2**

[3]** Every nontrivial tree with exactly k end-vertices has Steiner number k.

By Corollary 2.2 and Lemma 2.1, we have

** Corollary 2.3**

Let L(T) include all end-vertices of a tree T, then L(T) is a Steiner set of T.

Let $H=T[V-N[L(T)]]$ be the induced subgraph of $T$ from the set $V-N[L(T)]$. We have,

** Theorem 2.4**

For any nontrivial tree T, $\gamma_{st}(T)=|L(T)|+\gamma(H)$.

Proof.   Let $S$ be a minimum dominating set of $H$ and $\gamma(H)=|S|$. By Corollary 2.3, L(T) is a Steiner set of T. Hence the set $S\cup L(T)$ is a Steiner dominating set of T and $\gamma_{st}(T)\leq|L(T)|+\gamma(H)$.

Nextly, we prove $\gamma_{st}(T)\geq|L(T)|+\gamma(H)$. By contradiction, let $\gamma_{st}(T)<|L(T)|+\gamma(H)$ and there is a $\gamma_{st}$-set $S^{\prime}$ such that $\gamma_{st}(T)=|S^{\prime}|$. By Lemma 2.1, L(T) is a subset of each minimum Steiner set of $T$. Let $S^{\prime\prime}=S^{\prime}-L(T)$. By the definition of $H$, $S^{\prime\prime}$ is a minimum dominating set of $H$ such $|S^{\prime\prime}|=\gamma_{st}(T)-|L(T)|<\gamma(H)$, it is a contradiction. $\Box$

3 Linear algorithm for Forest Domination

In this section, we construct a linear algorithms for domination in forest. The algorithms is based on the algorithm for computing the domination number of an arbitrary tree by Mitchell, Cockayne and Hedetniemi [15].

By Theorem 2.4, the minimum Steinier dominating set of a tree is divided two subsets: $L(T)$ and the $\gamma$-set of subgraph $H$ of $T$.

By the definition of $H$, $H$ is a tree or a forest. So the algorithm in [15] has to be changed for computing the domination number of a forest. Algorithm 1 for domination of a forest $F$, and each tree $T$ in $F$ is rooted. Two linear arrays are maintained during this traversal process:

Parent[$i$]:contains the index of the parent of vertex $i$ in a forest $F$; in the Parent array, that the Parent of a vertex labelled $i$ is given by Parent[$i$], and Parent[$j$]=0 if vertex $j$ is the root of a tree in $F$; for any vertex labelled $i$ in $F$, Parent[$i$]$<i$.

Label[$i$]:contains three states:’Bound’,’Required’ and ’Free’; the usage of Label array is similar to the algorithm in [15].

Compared with the algorithm in [15], we add the condition that Parent[$i$] $\neq$ 0. This condition ensures that we construct the dominating set of each tree in $F$ by Algorithm 1 and get the minimum dominating set of a forest $F$.

** Theorem 3.1**

(Complexity of Algorithm 1). If the input forest to Algorithm 1 has order n, then both the space complexity and the worst-case time complexity of Algorithm 1 are O(n).

Proof.   Setp 1 can be performed in O(1) time. Steps 2-3, 4-11, 12-14 are three for-loops, and each operation in these loops can be performed in O(1) time. So the total operation time is $3n+1=O(n)$.

A total of $3n$ memory units are required to store the array Label,Parent and the set $D$. Two memory units are required to store the values of the variables $i$ and $N$. The space complexity of Algorithm 1 is therefore $3n+2=O(n)$. $\Box$

4 Linear algorithm for Tree Steiner Domination

In this section, we construct a linear algorithms for Steiner domination in a tree. By Theorem 2.4, the definition of $H$ and Algorithm 1, we only consider the structures of $L(T)$ and $H$. Five linear arrays are maintained during this traversal process:

Parent[$i$]:contains the index of the parent of vertex $i$ in tree $T$; in the Parent array, that the Parent of a vertex labelled $i$ is given by Parent[$i$], and Parent[$i$]=0 if vertex $i$ is the root of $T$; for any vertex labelled $i$ in $T$, Parent[$i$]$<i$.

Flag[$i$]:Flag[$i$]=0 if the vertex $i$ is a end-vertex of $T$, else Flag[$i$]=1.

PFlag[$i$]:PFlag[$i$]=1 if the vertex $i$ is adjacent to a end-vertex of $T$, else PFlag[$i$]=0.

Index[$i$]:contains the index in $T$ of the vertex $i$ in $H$.

NParent[$i$]:contains the index of the parent of vertex $i$ in a forest $H$; in the Parent array, that the Parent of a vertex labelled $i$ is given by Parent[$i$], and Parent[$j$]=0 if vertex $j$ is the root of a tree in $H$; for any vertex labelled $i$ in $H$, Parent[$i$]$<i$.

By the steps 1-23 in Algorithm 2, we get $L(T)$ (the end-vertex set of $T$) and NParent array of $H=G[V-N[L(T)]]$. We obtain the $\gamma$-set of $H$ by the step 24 in Algorithm 2 (Nparent array as a input of Algorithm 1). Finally, we have a minimum Steiner dominating set of tree $T$ by the step 25 in Algorithm 2.

We conclude this section with a result on the space and time complexities of Algorithm 2.

** Theorem 4.1**

(Complexity of Algorithm 2). If the input tree to Algorithm 2 has order n, then both the space complexity and the worst-case time complexity of Algorithm 2 are O(n).

Proof.   Setps 1 and 25 can be performed in O(1) time. Steps 2-4, 5-7, 8-10, 11-18, 19-23 are five for-loops, and each operation in these loops can be performed in O(1) time. So the total operation time of these loops is $4n+m$. The operation time in step 24 is $O(m)$ by Theorem 3.1. So the total operation time is $4n+m+2+O(m)=O(n)$.

A total of $8n$ memory units are required to store the array Label, Parent, NParent, Flag, PFlage, Index, the set $D$ and $SD$. Three memory units are required to store the values of the variables $i$ , $N$ and $m$. The space complexity of Algorithm 2 is therefore $8n+3=O(n)$. $\Box$