Finding Efficient Domination for $S_{1,1,5}$-Free Bipartite Graphs in Polynomial Time

TL;DR

This paper proves that the Efficient Domination problem, which is generally NP-complete, can be solved in polynomial time for bipartite graphs that do not contain an $S_{1,1,5}$ subgraph, expanding the classes of graphs with tractable solutions.

Contribution

The paper introduces a polynomial-time algorithm for solving the Efficient Domination problem specifically in $S_{1,1,5}$-free bipartite graphs, a previously unresolved class.

Findings

Efficient Domination problem is polynomial-time solvable for $S_{1,1,5}$-free bipartite graphs.

NP-completeness persists for many other bipartite graph classes.

The result extends the boundary of tractable cases for ED.

Abstract

A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (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 \NP-complete for various $H$-free bipartite graphs, e.g., Lu and Tang showed that ED is \NP-complete for chordal bipartite graphs and for planar bipartite graphs; actually, ED is \NP-complete even for planar bipartite graphs with vertex degree at most 3 and girth at least $g$ for every fixed $g$. Thus, ED is \NP-complete for $K_{1,4}$-free bipartite graphs and for $C_4$-free bipartite graphs. In this paper, we show that ED can be solved in polynomial time for $S_{1,1,5}$-free bipartite graphs.