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$m$-Eternal Dominating Set Problem on Subclasses of Chordal Graphs

Ashutosh Rai, Soumyashree Rana

TL;DR

The paper tackles the $m$-Eternal Dominating Set problem on chordal subclasses, establishing a dichotomy for split graphs: the problem is solvable in polynomial time for $K_{1,t}$-free split graphs with $t\le 4$ but NP-hard for $t\ge 5$, and proves NP-hardness on undirected path graphs. It further analyzes complexity differences among Dominating Set, Eternal Dominating Set, and $m$-Eternal Dominating Set via GP$_3$ and GP$_5$ graph constructions, showing linear-time solvability for DS and $m$-ED on GP$_3$ while Eternal is NP-hard, and showing DS is NP-hard while $m$-ED is poly-time on GP$_5$. The authors provide a detailed polynomial-time algorithm for $K_{1,4}$-free split graphs and a reduction-based NP-hardness proof for $K_{1,5}$-free splits, along with an NP-hardness result for undirected path graphs via 3D-matching. These results map a nuanced tractability landscape for dynamic domination variants on structured graphs and motivate future investigation into additional graph classes.

Abstract

A dominating set of a graph G(V, E) is a set of vertices D\subseteq V such that every vertex in V\D has a neighbor in D. An eternal dominating set extends this concept by placing mobile guards on the vertices of D. In response to an infinite sequence of attacks on unoccupied vertices, a guard can move to the attacked vertex from an adjacent position, ensuring that the new guards configuration remains a dominating set. In the one (all) guard(s) move model, only one (multiple) guard(s) moves(may move) per attack. The set of vertices representing the initial configuration of guards in one(all) guard move model is the eternal dominating set (m-eternal dominating set) of G. The minimum size of such a set in one(all) guard move model is called the eternal domination number (m-eternal domination number) of G, respectively. Given a graph G and an integer k, the m-Eternal Dominating Set asks whether G has an m-eternal dominating set of size at most k. In this work, we focus mainly on the computational complexity of m-Eternal Dominating Set in subclasses of chordal graphs. For split graphs, we show a dichotomy result by first designing a polynomial-time algorithm for K1,t-free split graphs with t\le 4, and then proving that the problem becomes NP-complete for t\ge 5. We showed that the problem is NP-hard on undirected path graphs. Moreover, we exhibit the computational complexity difference between the variants by showing the existence of two graph classes such that, in one, both Dominating Set and m-Eternal Dominating Set are solvable in polynomial time while Eternal Dominating Set is NP-hard, whereas in the other, Eternal Dominating Set is solvable in polynomial time and both Dominating Set and m-Eternal Dominating Set are NP-hard. Finally, we present a graph class where Dominating Set is NP-hard, but m-Eternal Dominating Set is efficiently solvable.

$m$-Eternal Dominating Set Problem on Subclasses of Chordal Graphs

TL;DR

The paper tackles the -Eternal Dominating Set problem on chordal subclasses, establishing a dichotomy for split graphs: the problem is solvable in polynomial time for -free split graphs with but NP-hard for , and proves NP-hardness on undirected path graphs. It further analyzes complexity differences among Dominating Set, Eternal Dominating Set, and -Eternal Dominating Set via GP and GP graph constructions, showing linear-time solvability for DS and -ED on GP while Eternal is NP-hard, and showing DS is NP-hard while -ED is poly-time on GP. The authors provide a detailed polynomial-time algorithm for -free split graphs and a reduction-based NP-hardness proof for -free splits, along with an NP-hardness result for undirected path graphs via 3D-matching. These results map a nuanced tractability landscape for dynamic domination variants on structured graphs and motivate future investigation into additional graph classes.

Abstract

A dominating set of a graph G(V, E) is a set of vertices D\subseteq V such that every vertex in V\D has a neighbor in D. An eternal dominating set extends this concept by placing mobile guards on the vertices of D. In response to an infinite sequence of attacks on unoccupied vertices, a guard can move to the attacked vertex from an adjacent position, ensuring that the new guards configuration remains a dominating set. In the one (all) guard(s) move model, only one (multiple) guard(s) moves(may move) per attack. The set of vertices representing the initial configuration of guards in one(all) guard move model is the eternal dominating set (m-eternal dominating set) of G. The minimum size of such a set in one(all) guard move model is called the eternal domination number (m-eternal domination number) of G, respectively. Given a graph G and an integer k, the m-Eternal Dominating Set asks whether G has an m-eternal dominating set of size at most k. In this work, we focus mainly on the computational complexity of m-Eternal Dominating Set in subclasses of chordal graphs. For split graphs, we show a dichotomy result by first designing a polynomial-time algorithm for K1,t-free split graphs with t\le 4, and then proving that the problem becomes NP-complete for t\ge 5. We showed that the problem is NP-hard on undirected path graphs. Moreover, we exhibit the computational complexity difference between the variants by showing the existence of two graph classes such that, in one, both Dominating Set and m-Eternal Dominating Set are solvable in polynomial time while Eternal Dominating Set is NP-hard, whereas in the other, Eternal Dominating Set is solvable in polynomial time and both Dominating Set and m-Eternal Dominating Set are NP-hard. Finally, we present a graph class where Dominating Set is NP-hard, but m-Eternal Dominating Set is efficiently solvable.
Paper Structure (12 sections, 19 theorems, 1 figure, 1 table)

This paper contains 12 sections, 19 theorems, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Our work
  4. Preliminaries
  5. Complexity Difference
  6. Dominating Set vs. $m$-Eternal Dominating Set
  7. Complexity Dichotomy for Split Graphs
  8. $K_{1,3}$-free Split Graphs
  9. $K_{1,4}$-free Split Graphs
  10. $K_{1,5}$-free Split Graphs
  11. Undirected Path Graph
  12. Conclusion

Key Result

Theorem 1

gavril1974intersection A graph $G=(V, E)$ is an undirected path graph if and only if there exists a tree $T$ (called as clique tree of $G$) with $V(T)=\mathcal{C}(G)$, where $\mathcal{C}(G)$ is the set of all maximal cliques of $G$, such that $T[\mathcal{C}_v(G)]$ is a path in $T$ for each $v\in V(G

Figures (1)

  • Figure 1: In the hierarchy diagram of chordal graph classes, an edge from one class to another indicates that the upper class is a superclass of the lower one. Each class is annotated with a colored box: red denotes that $m$-Eternal Dominating Set is NP-hard, green denotes polynomial-time solvability, and black indicates that the complexity remains unknown.

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Theorem 5
  • Lemma 6
  • Definition 11
  • Lemma 12
  • Lemma 13
  • Corollary 14
  • Corollary 15
  • ...and 15 more