On the Enumeration of Minimal Dominating Sets and Related Notions

TL;DR

This paper explores the enumeration of minimal dominating sets in graphs, establishing polynomial reductions with hypergraph transversals, and provides efficient algorithms for specific graph classes, advancing understanding of related enumeration problems.

Contribution

It demonstrates polynomial equivalences between minimal dominating set enumeration and hypergraph transversals, and develops output-polynomial algorithms for certain graph classes.

Findings

Polynomial reduction between Dom-Enum and Trans-Enum problems.

Output-polynomial algorithms for split graphs and P6-free chordal graphs.

Equivalence of enumeration complexities for various dominating set problems.

Abstract

A dominating set $D$ in a graph is a subset of its vertex set such that each vertex is either in $D$ or has a neighbour in $D$. In this paper, we are interested in the enumeration of (inclusion-wise) minimal dominating sets in graphs, called the Dom-Enum problem. It is well known that this problem can be polynomially reduced to the Trans-Enum problem in hypergraphs, i.e., the problem of enumerating all minimal transversals in a hypergraph. Firstly we show that the Trans-Enum problem can be polynomially reduced to the Dom-Enum problem. As a consequence there exists an output-polynomial time algorithm for the Trans-Enum problem if and only if there exists one for the Dom-Enum problem. Secondly, we study the Dom-Enum problem in some graph classes. We give an output-polynomial time algorithm for the Dom-Enum problem in split graphs, and introduce the completion of a graph to obtain an output-polynomial time algorithm for the Dom-Enum problem in $P_6$-free chordal graphs, a proper superclass of split graphs. Finally, we investigate the complexity of the enumeration of (inclusion-wise) minimal connected dominating sets and minimal total dominating sets of graphs. We show that there exists an output-polynomial time algorithm for the Dom-Enum problem (or equivalently Trans-Enum problem) if and only if there exists one for the following enumeration problems: minimal total dominating sets, minimal total dominating sets in split graphs, minimal connected dominating sets in split graphs, minimal dominating sets in co-bipartite graphs.