TY - JOUR

T1 - Critical Sets in Bipartite Graphs

AU - Levit, Vadim E.

AU - Mandrescu, Eugen

PY - 2013/9

Y1 - 2013/9

N2 - Let G = (V, E) be a graph. A set I ⊆ V is independent if no two vertices from I are adjacent, and by Ind(G) we mean the family of all independent sets of G, while core(G) is the intersection of all maximum independent sets [4]. The number dc(G) = max{{pipe}I{pipe} - {pipe}N(I){pipe}: I ∈ Ind(G)} is called the critical difference of G. A set X is critical if {pipe}X{pipe} - {pipe}N(X){pipe} = dc(G) [10]. For a bipartite graph G = (A, B, E), Ore [7] defined δ0(A) = max{{pipe}X{pipe} - {pipe}N(X){pipe}: X ⊆ A}. In this paper, we prove that dc(G) = δ0(A)+δ0(B) and ker(G) = core(G) hold for every bipartite graph G = (A, B, E), where ker(G) denotes the intersection of all critical independent sets.

AB - Let G = (V, E) be a graph. A set I ⊆ V is independent if no two vertices from I are adjacent, and by Ind(G) we mean the family of all independent sets of G, while core(G) is the intersection of all maximum independent sets [4]. The number dc(G) = max{{pipe}I{pipe} - {pipe}N(I){pipe}: I ∈ Ind(G)} is called the critical difference of G. A set X is critical if {pipe}X{pipe} - {pipe}N(X){pipe} = dc(G) [10]. For a bipartite graph G = (A, B, E), Ore [7] defined δ0(A) = max{{pipe}X{pipe} - {pipe}N(X){pipe}: X ⊆ A}. In this paper, we prove that dc(G) = δ0(A)+δ0(B) and ker(G) = core(G) hold for every bipartite graph G = (A, B, E), where ker(G) denotes the intersection of all critical independent sets.

KW - critical set

KW - independent set

KW - matching

UR - http://www.scopus.com/inward/record.url?scp=84883778943&partnerID=8YFLogxK

U2 - 10.1007/s00026-013-0195-4

DO - 10.1007/s00026-013-0195-4

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AN - SCOPUS:84883778943

SN - 0218-0006

VL - 17

SP - 543

EP - 548

JO - Annals of Combinatorics

JF - Annals of Combinatorics

IS - 3

ER -