Polynomial-time Algorithms for Weighted Efficient Domination Problems in AT-free Graphs and Dually Chordal Graphs

TL;DR

This paper introduces polynomial-time algorithms for solving weighted efficient domination problems in AT-free and dually chordal graphs by reducing them to maximum weight independent set problems, advancing the understanding of graph domination complexities.

Contribution

The paper presents a novel framework that reduces weighted efficient domination problems to maximum weight independent set problems, providing polynomial algorithms for specific graph classes.

Findings

Polynomial-time algorithms for AT-free graphs

Polynomial-time algorithms for dually chordal graphs

Resolution of an open problem in strongly chordal graphs

Abstract

An efficient dominating set (or perfect code) in a graph is a set of vertices the closed neighborhoods of which partition the vertex set of the graph. The minimum weight efficient domination problem is the problem of finding an efficient dominating set of minimum weight in a given vertex-weighted graph; the maximum weight efficient domination problem is defined similarly. We develop a framework for solving the weighted efficient domination problems based on a reduction to the maximum weight independent set problem in the square of the input graph. Using this approach, we improve on several previous results from the literature by deriving polynomial-time algorithms for the weighted efficient domination problems in the classes of dually chordal and AT-free graphs. In particular, this answers a question by Lu and Tang regarding the complexity of the minimum weight efficient domination problem in strongly chordal graphs.