TY - JOUR

T1 - On α-critical edges in König-Egerváry graphs

AU - Levit, Vadim E.

AU - Mandrescu, Eugen

PY - 2006/8/6

Y1 - 2006/8/6

N2 - The stability number of a graph G, denoted by α (G), is the cardinality of a stable set of maximum size in G. If α (G - e) > α (G), then e is an α-critical edge, and if μ (G - e) < μ (G), then e is a μ-critical edge, where μ (G) is the cardinality of a maximum matching in G. G is a König-Egerváry graph if its order equals α (G) + μ (G). Beineke, Harary and Plummer have shown that the set of α-critical edges of a bipartite graph forms a matching. In this paper we generalize this statement to König-Egerváry graphs. We also prove that in a König-Egerváry graph α-critical edges are also μ-critical, and that they coincide in bipartite graphs. For König-Egerváry graphs, we characterize μ-critical edges that are also α-critical. Eventually, we deduce that α (T) = ξ (T) + η (T) holds for any tree T, and describe the König-Egerváry graphs enjoying this property, where ξ (G) is the number of α-critical vertices and η (G) is the number of α-critical edges.

AB - The stability number of a graph G, denoted by α (G), is the cardinality of a stable set of maximum size in G. If α (G - e) > α (G), then e is an α-critical edge, and if μ (G - e) < μ (G), then e is a μ-critical edge, where μ (G) is the cardinality of a maximum matching in G. G is a König-Egerváry graph if its order equals α (G) + μ (G). Beineke, Harary and Plummer have shown that the set of α-critical edges of a bipartite graph forms a matching. In this paper we generalize this statement to König-Egerváry graphs. We also prove that in a König-Egerváry graph α-critical edges are also μ-critical, and that they coincide in bipartite graphs. For König-Egerváry graphs, we characterize μ-critical edges that are also α-critical. Eventually, we deduce that α (T) = ξ (T) + η (T) holds for any tree T, and describe the König-Egerváry graphs enjoying this property, where ξ (G) is the number of α-critical vertices and η (G) is the number of α-critical edges.

KW - Bipartite graph

KW - Core

KW - Critical edge

KW - Maximum matching

KW - Maximum stable set

KW - Tree

UR - http://www.scopus.com/inward/record.url?scp=33745962997&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2006.05.001

DO - 10.1016/j.disc.2006.05.001

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AN - SCOPUS:33745962997

SN - 0012-365X

VL - 306

SP - 1684

EP - 1693

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 15

ER -