Enumeration and Maximum Number of Minimal Connected Vertex Covers in Graphs

TL;DR

This paper investigates the maximum number of minimal connected vertex covers in graphs, providing upper bounds, enumeration algorithms, and tight bounds for specific graph classes, advancing understanding of this classical problem.

Contribution

It introduces the first bounds on the maximum number of minimal connected vertex covers and offers enumeration algorithms, with improved bounds for special graph classes.

Findings

Maximum number of minimal connected vertex covers is at most 1.8668^n.

Enumeration of all minimal connected vertex covers can be done in O(1.8668^n) time.

Tight bounds are established for chordal and distance-hereditary graphs.

Abstract

Connected Vertex Cover is one of the classical problems of computer science, already mentioned in the monograph of Garey and Johnson. Although the optimization and decision variants of finding connected vertex covers of minimum size or weight are well studied, surprisingly there is no work on the enumeration or maximum number of minimal connected vertex covers of a graph. In this paper we show that the maximum number of minimal connected vertex covers of a graph is at most 1.8668^n, and these can be enumerated in time O(1.8668^n). For graphs of chordality at most 5, we are able to give a better upper bound, and for chordal graphs and distance-hereditary graphs we are able to give tight bounds on the maximum number of minimal connected vertex covers.