On Efficient Domination for Some Classes of $H$-Free Bipartite Graphs

TL;DR

This paper investigates the computational complexity of the Efficient Domination problem in specific classes of bipartite graphs, providing polynomial-time algorithms for some classes and proving NP-completeness for others.

Contribution

It establishes polynomial-time solvability of ED for certain $H$-free bipartite graphs and proves NP-completeness for bipartite graphs with small diameter.

Findings

Polynomial-time algorithms for ED in $P_7$-free and $ ext{ell} P_4$-free bipartite graphs.

Polynomial-time solvability of ED in $P_9$-free bipartite graphs with degree constraints.

NP-completeness of ED in bipartite graphs with diameter at most 6.

Abstract

A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (\emph{e.d.s.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which asks for the existence of an e.d.s.\ in $G$, is known to be \NP-complete even for very restricted $H$-free graph classes such as for $2P_3$-free chordal graphs while it is solvable in polynomial time for $P_6$-free graphs. Here we focus on $H$-free bipartite graphs: We show that (weighted) ED can be solved in polynomial time for $H$-free bipartite graphs when $H$ is $P_7$ or $\ell P_4$ for fixed $\ell$, and similarly for $P_9$-free bipartite graphs with vertex degree at most 3, and when $H$ is $S_{2,2,4}$. Moreover, we show that ED is \NP-complete for bipartite graphs with diameter at most 6.