Bisplit graphs -- A Structural and algorithmic study

TL;DR

This paper investigates the computational complexity of the secure domination problem in various graph classes, establishing NP-completeness in some cases and polynomial solvability in others, with implications for approximation limits.

Contribution

It provides a detailed complexity analysis of the secure domination problem on bisplit graphs and related classes, including NP-completeness, polynomial cases, and approximation bounds.

Findings

NP-complete on star, convex split, and bisplit graphs

Polynomial-time solvable in chain graphs

Hard to approximate within a logarithmic factor

Abstract

A dominating set $S$ of a graph $G(V,E)$ is called a \textit{secure dominating set} if each vertex $u \in V(G) \setminus S$ is adjacent to a vertex $v \in S$ such that $(S \setminus \{v\}) \cup \{u\}$ is a dominating set of $G$. The \textit{secure domination number} $\gamma_s(G)$ of $G$ is the minimum cardinality of a secure dominating set of $G$. The \textit{Minimum Secure Domination problem} is to find a secure dominating set of a graph $G$ of cardinality $\gamma_s(G)$. In this paper, the computational complexity of the secure domination problem on several graph classes is investigated. The decision version of secure domination problem was shown to be NP-complete on star(comb) convex split graphs and bisplit graphs. So we further focus on complexity analysis of secure domination problem under additional structural restrictions on bisplit graphs. In particular, by imposing chordality as a parameter, we analyse its impact on the computational status of the problem on bisplit graphs. We establish the P versus NP-C dichotomy status of secure domination problem under restrictions on cycle length within bisplit graphs. In addition, we establish that the problem is polynomial-time solvable in chain graphs. We also prove that the secure domination problem cannot be approximated for a bisplit graph within a factor of $(1-\epsilon)~ln~|V|$ for any $\epsilon > 0$, unless $NP \subseteq DTIME(|V|^{O(log~log~|V|)})$.