Subgraph characterization of red/blue-split graph and kőnig egerváry graphs

A matching M is uniquely restricted in a graph G if its saturated vertices induce a subgraph which has a unique perfect matching, namely M itself [M.C. Golumbic, T. Hirst, M. Lewenstein, Uniquely restricted matchings, Algorithmica 31 (2001) 139-154]. G is a König-Egerváry graph provided (G) + (G) = |V (G)| [R.W. Deming, Independence numbers of graphs-an extension of the König-Egerváry theorem, Discrete Math. 27 (1979) 23-33; F. Sterboul, A characterization of the graphs in which the transversal number equals the matching number, J. Combin. Theory Ser. B 27 (1979) 228-229], where (G) is the size of a maximum matching and (G) is the cardinality of a maximum stable set. S is a local maximum stable set of G, and we write S ∈ (G), if S is a maximum stable set of the subgraph spanned by S ∪ N(S), where N(S) is the neighborhood of S. Nemhauser and Trotter [Vertex packings: structural properties and algorithms, Math. Programming 8 (1975) 232-248], proved that any S ∈ (G) is a subset of a maximum stable set of G. In [V.E. Levit, E. Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math. 132 (2003) 163-174] we have proved that for a bipartite graph G, (G) is a greedoid on its vertex set if and only if all its maximum matchings are uniquely restricted. In this paper we demonstrate that if G is a triangle-free graph, then (G) is a greedoid if and only if all its maximum matchings are uniquely restricted and for any S ∈ (G), the subgraph spanned by S ∪ N(S) is a König-Egerváry graph.

show abstract

“…Properties of König-Egerváry graphs were discussed in a number of papers, e.g. [2,10,11,15,[19][20][21]25].…”

Section: Theorem 11 (Nemhauser and Trottermentioning

confidence: 99%

Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids

Levit

Mândrescu

2007

Discrete Applied Mathematics

View full text Add to dashboard Cite

show abstract

“…Lovász [20] used the theory of matching-covered graphs to give an excluded-subgraph characterization of König graphs that contain a perfect matching. Korach et al [19] generalized this and gave an excluded-subgraph characterization for the class of all König graphs.…”

Section: Introductionmentioning

confidence: 99%

“…König graphs have been studied for a fairly long time from a structural point of view [2,6,19,20,32]. Both Deming [6] and Sterboul [32] gave independent characterizations of König graphs and showed that König graphs can be recognized in polynomial time.…”

Section: Introductionmentioning

confidence: 99%

The Complexity of König Subgraph Problems and Above-Guarantee Vertex Cover

et al. 2010

View full text Add to dashboard Cite

Abstract. A graph is König-Egerváry if the size of a minimum vertex cover equals that of a maximum matching in the graph. These graphs have been studied extensively from a graph-theoretic point of view. In this paper, we introduce and study the algorithmic complexity of finding König-Egerváry subgraphs of a given graph. In particular, given a graph G and a nonnegative integer k, we are interested in the following questions:1. does there exist a set of k vertices (edges) whose deletion makes the graph König-Egerváry? 2. does there exist a set of k vertices (edges) that induce a König-Egerváry subgraph?We show that these problems are NP-complete and study their complexity from the points of view of approximation and parameterized complexity. Towards this end, we first study the algorithmic complexity of Above Guarantee Vertex Cover, where one is interested in minimizing the additional number of vertices needed beyond the maximum matching size for the vertex cover. Further, while studying the parameterized complexity of the problem of deleting k vertices to obtain a König-Egerváry graph, we show a number of interesting structural results on matchings and vertex covers which could be useful in other contexts.

show abstract

“…In this paper, we study graphs G with the property ν(G) = τ f (G). The class of such graphs, known as stable graphs, subsumes the well-studied class of König-Egerváry (KEG) graphs (e.g., see [20,13,14,15]) for which ν(G) = τ (G). Stable graphs arise quite naturally in the study of cooperative matching games introduced by Shapley and Shubik in their seminal paper [19].…”

Section: Introductionmentioning

confidence: 99%

Finding small stabilizers for unstable graphs

et al. 2014

Self Cite

View full text Add to dashboard Cite

Abstract. An undirected graph G = (V, E) is stable if its inessential vertices (those that are exposed by at least one maximum matching) form a stable set. We call a set of edges F ⊆ E a stabilizer if its removal from G yields a stable graph. In this paper we study the following natural edgedeletion question: given a graph G = (V, E), can we find a minimumcardinality stabilizer? Stable graphs play an important role in cooperative game theory. In the classic matching game introduced by Shapley and Shubik [19] we are given an undirected graph G = (V, E) where vertices represent players, and we define the value of each subset S ⊆ V as the cardinality of a maximum matching in the subgraph induced by S. The core of such a game contains all fair allocations of the value of V among the players, and is well-known to be non-empty iff graph G is stable. The stabilizer problem addresses the question of how to modify the graph to ensure that the core is non-empty. We show that this problem is vertex-cover hard. We then prove that there is a minimum-cardinality stabilizer that avoids some maximum matching of G. We use this insight to give efficient approximation algorithms for sparse graphs and for regular graphs.

show abstract

Subgraph characterization of red/blue-split graph and kőnig egerváry graphs

Cited by 29 publications

References 0 publications

Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids

Triangle-free graphs with uniquely restricted maximum matchings and their corresponding greedoids

The Complexity of König Subgraph Problems and Above-Guarantee Vertex Cover

Finding small stabilizers for unstable graphs

Contact Info

Product

Resources

About