Efficient and Perfect domination on circular-arc graphs

TL;DR

This paper presents efficient algorithms for finding perfect and efficient dominating sets in circular-arc graphs, addressing NP-hard problems with practical solutions for this specific graph family.

Contribution

The paper introduces the first polynomial-time algorithms for perfect and efficient domination problems on circular-arc graphs.

Findings

Efficient algorithms for perfect domination in circular-arc graphs.

Efficient algorithms for efficient domination in circular-arc graphs.

Addresses NP-hardness by providing practical solutions for this graph class.

Abstract

Given a graph $G = (V,E)$, a \emph{perfect dominating set} is a subset of vertices $V' \subseteq V(G)$ such that each vertex $v \in V(G)\setminus V'$ is dominated by exactly one vertex $v' \in V'$. An \emph{efficient dominating set} is a perfect dominating set $V'$ where $V'$ is also an independent set. These problems are usually posed in terms of edges instead of vertices. Both problems, either for the vertex or edge variant, remains NP-Hard, even when restricted to certain graphs families. We study both variants of the problems for the circular-arc graphs, and show efficient algorithms for all of them.