A characterization of König-Egerváry graphs with extendable vertex covers

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Highlights

  • We characterize König-Egerváry graphs with a minimum vertex cover containing a specified subset of vertices.

  • We also completely characterize all König-Egerváry graphs with a unique minimum vertex cover.

  • Our characterizations are algorithmic and can be checked efficiently.

Abstract

It is well known that in a bipartite (and more generally in a König-Egerváry) graph, the size of the minimum vertex cover is equal to the size of the maximum matching. We first address the question whether (and if not, when) this property still holds in a König-Egerváry graph if we consider vertex covers containing a given subset of vertices. We characterize such graphs using the classic notions of alternating paths and flowers used in Edmonds' matching algorithm. We then use the notions of alternating paths and flowers in König-Egerváry graphs to give a complete characterization of such graphs that have a unique minimum vertex cover.

Introduction

The classic notions of matchings and vertex covers have been extensively studied for several decades in the area of Combinatorial Optimization. In 1931, König and Egerváry independently proved a result of fundamental importance: in a bipartite graph the size of a maximum matching equals that of a minimum vertex cover [4]. This led to a polynomial-time algorithm for finding a minimum vertex cover in bipartite graphs. Interestingly, this min-max relationship holds for a larger class of graphs known as König-Egerváry graphs that includes the set of all bipartite graphs as a proper subclass. Our first result in this paper is an extension of this classic result. That is, we address the following question:

When does a König-Egerváry graph have a minimum vertex cover containing a specified subset of vertices in the graph?

We resolve this problem by obtaining an excluded-subgraph characterization for König-Egerváry graphs satisfying this property. Specifically, let G be a König-Egerváry graph, let M be a maximum matching of G and let S be a set of vertices containing exactly one vertex from some of the edges of M. Then G has a minimum vertex cover of size |M| containing S if and only if it does not contain certain kinds of alternating paths or flowers with respect to M. The notions of alternating paths and flowers are the same as the one used in the classic maximum matching algorithm of Edmonds [2] on general graphs.

Our characterization implies a polynomial-time algorithm to test whether a König-Egerváry graph has a minimum vertex cover containing a given subset of vertices. This algorithm can be viewed as a generalization of the classic polynomial-time algorithm to find a minimum vertex cover of a given König-Egerváry graph.

Finally, using insights from the above characterization, we also give a complete characterization of König-Egerváry graphs with a unique minimum vertex cover. This particular question has been considered by Levit and Mandrescu [3].

Section snippets

Preliminaries

In what follows, we fix a simple graph G=(V,E) and a matching M of G. We use m and n to denote the number of edges and vertices of G respectively.

Definition 1

A sequence of distinct vertices v1,,vt is said to be a path in G if either t=1 or for every i{1,,t1}, (vi,vi+1) is an edge in G. Similarly, a sequence of vertices v1,,vt,v1 is said to be a cycle in G if t3, v1,,vt is a path in G and (vt,v1) is an edge in G. The length of a path (cycle) is the number of edges in the path (respectively, cycle).

Structure of König-Egerváry graphs with extendable vertex covers

In this section we obtain an extension of the classic result of König and Egerváry relating maximum matchings and minimum vertex covers. More precisely, we address the following question.

When does a König-Egerváry graph have a minimum vertex cover containing a specified subset of vertices of the graph?

While the question can be considered to be an extension of that studied by König-Egerváry and Egerváry, the answer we obtain revolves around the structures of alternating paths and flowers (see

Computing extendable vertex covers

This section is devoted to an algorithmic version of the results in the previous section. We show that one can compute a minimum vertex cover containing the annotated vertices of a König-Egerváry graph (if one exists) in polynomial time. To be specific, our goal is to design a polynomial-time algorithm that, given a König-Egerváry graph G=(A,B,E), LA and RB, correctly decides whether G has a minimum vertex cover containing LR and if the answer is affirmative then computes one such vertex

Characterizing König-Egerváry graphs with a unique minimum vertex cover

In this section, we give a complete characterization of König-Egerváry graphs with a unique minimum vertex cover or equivalently, those with a unique maximum independent set. The following observation is a consequence of Lemma 4 and implies that it is sufficient for us to restrict our attention to König-Egerváry graphs with perfect matchings.

Observation 4

Consider a König-Egerváry graph G=(A,B,E) with a maximum matching M where U is the set of vertices left unsaturated by M. Then, G has a unique minimum

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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The results in Section 3 of this paper were part of the article [6] that appeared in the proceedings of ESA 2011.

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