TY - JOUR

T1 - On the structure of the minimum critical independent set of a graph

AU - Levit, Vadim E.

AU - Mandrescu, Eugen

PY - 2013

Y1 - 2013

N2 - Let G=(V,E). A set S⊆V is independent if no two vertices from S are adjacent, and by Ind(G) we mean the family of all the independent sets of G. The number d(X)=X-N(X) is the difference of X⊆V, and A⋯Ind(G) is critical if d(A)=maxd(I):I⋯Ind(G) [18]. Let us recall the following definitions: core(G)=S:S is a maximum independent set[10],ker(G)=S:S is a critical independent set [12]. Recently, it was established that ker(G)⊆core(G) is true for every graph [12], while the corresponding equality holds for bipartite graphs [13]. In this paper, we present various structural properties of ker(G). The main finding claims that ker(G)= S0:S0 is an inclusion minimal independent set with d( S0)=1=S0:S0 is an inclusion minimal independent set with d(S0)>0.

AB - Let G=(V,E). A set S⊆V is independent if no two vertices from S are adjacent, and by Ind(G) we mean the family of all the independent sets of G. The number d(X)=X-N(X) is the difference of X⊆V, and A⋯Ind(G) is critical if d(A)=maxd(I):I⋯Ind(G) [18]. Let us recall the following definitions: core(G)=S:S is a maximum independent set[10],ker(G)=S:S is a critical independent set [12]. Recently, it was established that ker(G)⊆core(G) is true for every graph [12], while the corresponding equality holds for bipartite graphs [13]. In this paper, we present various structural properties of ker(G). The main finding claims that ker(G)= S0:S0 is an inclusion minimal independent set with d( S0)=1=S0:S0 is an inclusion minimal independent set with d(S0)>0.

KW - Core

KW - Critical set

KW - Independent set

KW - Ker

KW - Matching

UR - http://www.scopus.com/inward/record.url?scp=84872105432&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2012.12.008

DO - 10.1016/j.disc.2012.12.008

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AN - SCOPUS:84872105432

SN - 0012-365X

VL - 313

SP - 605

EP - 610

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 5

ER -