Generating All Minimal Edge Dominating Sets with Incremental-Polynomial Delay

TL;DR

This paper presents algorithms for efficiently enumerating all minimal edge dominating sets in graphs, with improved delay and running time, especially for bipartite graphs, advancing the field of output-sensitive enumeration.

Contribution

The paper introduces new algorithms with incremental-polynomial delay for enumerating minimal edge dominating sets, including specialized results for bipartite graphs, improving upon previous methods.

Findings

Algorithms achieve incremental-polynomial delay O(m^5 |L|) and O(m^4 |L|).

Enumeration is output-polynomial time for line graphs and bipartite line graphs.

Results relate to minimal transversals enumeration in hypergraphs.

Abstract

For an arbitrary undirected simple graph G with m edges, we give an algorithm with running time O(m^4 |L|^2) to generate the set L of all minimal edge dominating sets of G. For bipartite graphs we obtain a better result; we show that their minimal edge dominating sets can be enumerated in time O(m^4 |L|). In fact our results are stronger; both algorithms generate the next minimal edge dominating set with incremental-polynomial delay O(m^5 |L|) and O(m^4 |L|) respectively, when L is the set of already generated minimal edge dominating sets. Our algorithms are tailored for and solve the equivalent problems of enumerating minimal (vertex) dominating sets of line graphs and line graphs of bipartite graphs, with incremental-polynomial delay, and consequently in output-polynomial time. Enumeration of minimal dominating sets in graphs has very recently been shown to be equivalent to enumeration of minimal transversals in hypergraphs. The question whether the minimal transversals of a hypergraph can be enumerated in output-polynomial time is a fundamental and challenging question in Output-Sensitive Enumeration; it has been open for several decades and has triggered extensive research in the field.