TY - JOUR

T1 - Critical Independent Sets and König-Egerváry Graphs

AU - Levit, Vadim E.

AU - Mandrescu, Eugen

PY - 2012/3

Y1 - 2012/3

N2 - A set S of vertices is independent or stable in a graph G, and we write S ∈ Ind (G), if no two vertices from S are adjacent, and α(G) is the cardinality of an independent set of maximum size, while core(G) denotes the intersection of all maximum independent sets. G is called a König-Egerváry graph if its order equals α(G) + μ(G), where μ(G) denotes the size of a maximum matching. The number def (G) = {pipe}V(G){pipe} -2μ(G) is the deficiency of G. The number def(G)={pipe}V(G){pipe}-2μ(G) is the deficiency of G. The number d(G)=max{{pipe}S{pipe}-{pipe}N(A){pipe}=d(G)}, where N(S) is the neighbourhood of S, and α c(G) denotes the maximum size of a critical independent set. Lrson (Eur J Comb 32:294-300, 2011)demonstrated that G is a k̈nig-Egerváry graph if and only if there exists a maximum independent set that is also critical, i.e., α c(G)=α(G). In this paper we prove that: (i) d(G)={pipe}(G){pipe}-{pipe}N(core(G)){pipe}=α(G)=def(G) holds for every König-Egerváry graph G; (ii) G is König-Egerváry graph if and only if each maximum independent set of G is critical.

AB - A set S of vertices is independent or stable in a graph G, and we write S ∈ Ind (G), if no two vertices from S are adjacent, and α(G) is the cardinality of an independent set of maximum size, while core(G) denotes the intersection of all maximum independent sets. G is called a König-Egerváry graph if its order equals α(G) + μ(G), where μ(G) denotes the size of a maximum matching. The number def (G) = {pipe}V(G){pipe} -2μ(G) is the deficiency of G. The number def(G)={pipe}V(G){pipe}-2μ(G) is the deficiency of G. The number d(G)=max{{pipe}S{pipe}-{pipe}N(A){pipe}=d(G)}, where N(S) is the neighbourhood of S, and α c(G) denotes the maximum size of a critical independent set. Lrson (Eur J Comb 32:294-300, 2011)demonstrated that G is a k̈nig-Egerváry graph if and only if there exists a maximum independent set that is also critical, i.e., α c(G)=α(G). In this paper we prove that: (i) d(G)={pipe}(G){pipe}-{pipe}N(core(G)){pipe}=α(G)=def(G) holds for every König-Egerváry graph G; (ii) G is König-Egerváry graph if and only if each maximum independent set of G is critical.

KW - Core

KW - Critical difference

KW - Critical independent set

KW - Deficiency

KW - Maximum independent set

KW - Maximum matching

UR - http://www.scopus.com/inward/record.url?scp=84857653881&partnerID=8YFLogxK

U2 - 10.1007/s00373-011-1037-y

DO - 10.1007/s00373-011-1037-y

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AN - SCOPUS:84857653881

SN - 0911-0119

VL - 28

SP - 243

EP - 250

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

IS - 2

ER -