Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2602.02135

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

Ashutosh Rai·Soumyashree Rana
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.CCcs.DM

Keep this discovery

BibTeXRIS

Ashutosh Rai, Soumyashree Rana. 2026-02-02. $m$-Eternal Dominating Set Problem on Subclasses of Chordal Graphs. https://arxiv.org/abs/2602.02135

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗
arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗