Efficient Dominating and Edge Dominating Sets for Graphs and Hypergraphs

TL;DR

This paper studies efficient dominating and edge dominating sets in graphs and hypergraphs, providing complexity results and linear time algorithms for specific graph classes, and extending the problems to hypergraphs with varied complexity.

Contribution

It offers a unified framework for these problems, solves open questions, and presents new polynomial and linear time algorithms for certain graph classes and hypergraphs.

Findings

Linear time algorithms for ED and EED on dually chordal graphs

NP-completeness of ED on alpha-acyclic hypergraphs

Polynomial-time solutions for ED and EED on hypertrees

Abstract

Let G=(V,E) be a graph. A vertex dominates itself and all its neighbors, i.e., every vertex v in V dominates its closed neighborhood N[v]. A vertex set D in G is an efficient dominating (e.d.) set for G if for every vertex v in V, there is exactly one d in D dominating v. An edge set M is an efficient edge dominating (e.e.d.) set for G if it is an efficient dominating set in the line graph L(G) of G. The ED problem (EED problem, respectively) asks for the existence of an e.d. set (e.e.d. set, respectively) in the given graph. We give a unified framework for investigating the complexity of these problems on various classes of graphs. In particular, we solve some open problems and give linear time algorithms for ED and EED on dually chordal graphs. We extend the two problems to hypergraphs and show that ED remains NP-complete on alpha-acyclic hypergraphs, and is solvable in polynomial time on hypertrees, while EED is polynomial on alpha-acyclic hypergraphs and NP-complete on hypertrees.