New Polynomial Cases of the Weighted Efficient Domination Problem

TL;DR

This paper identifies new polynomial-time solvable cases of the weighted efficient domination problem in certain graph classes, expanding the understanding of its computational complexity.

Contribution

It demonstrates polynomial algorithms for WED on various subclasses of 2P3-free and P7-free graphs, and finds minimum weight e.d. with degree constraints.

Findings

WED is polynomial-time solvable on (P2+P4)-free graphs

WED is polynomial-time solvable on P5-free graphs

Minimum weight e.d. with degree ≤ 2 can be found in polynomial time

Abstract

Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete even for very restricted graph classes. In particular, the ED problem remains NP-complete for 2P3-free graphs and thus for P7-free graphs. We show that the weighted version of the problem (abbreviated WED) is solvable in polynomial time on various subclasses of 2P3-free and P7-free graphs, including (P2+P4)-free graphs, P5-free graphs and other classes. Furthermore, we show that a minimum weight e.d. consisting only of vertices of degree at most 2 (if one exists) can be found in polynomial time. This contrasts with our NP-completeness result for the ED problem on planar bipartite graphs with maximum degree 3.