TY - JOUR

T1 - W2-graphs and shedding vertices.

AU - Levit, Vadim E.

AU - Mandrescu, Eugen

N1 - DBLP License: DBLP's bibliographic metadata records provided through http://dblp.org/ are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.

PY - 2017/8

Y1 - 2017/8

N2 - A graph is well-covered if all its maximal independent sets are of the same size (Plummer, 1970). A well-covered graph is 1-well-covered if the deletion of every vertex leaves a graph which is well-covered as well (Staples, 1975). A graph G belongs to class Wn if every n pairwise disjoint independent sets in G are included in n pairwise disjoint maximum independent sets (Staples, 1975). Clearly, W1 is the family of all well-covered graphs. It turns out that G∈W2 if and only if it is a 1-well-covered graph without isolated vertices. We show that deleting a shedding vertex does not change the maximum size of a maximal independent set including a given A∈Ind(G) in a graph G, where Ind(G) is the family of all the independent sets. Specifically, for well-covered graphs, it means that the vertex v is shedding if and only if G−v is well-covered.

AB - A graph is well-covered if all its maximal independent sets are of the same size (Plummer, 1970). A well-covered graph is 1-well-covered if the deletion of every vertex leaves a graph which is well-covered as well (Staples, 1975). A graph G belongs to class Wn if every n pairwise disjoint independent sets in G are included in n pairwise disjoint maximum independent sets (Staples, 1975). Clearly, W1 is the family of all well-covered graphs. It turns out that G∈W2 if and only if it is a 1-well-covered graph without isolated vertices. We show that deleting a shedding vertex does not change the maximum size of a maximal independent set including a given A∈Ind(G) in a graph G, where Ind(G) is the family of all the independent sets. Specifically, for well-covered graphs, it means that the vertex v is shedding if and only if G−v is well-covered.

KW - 1-well-covered graph

KW - differential of a set

KW - matching

KW - maximum independent set

KW - shedding vertex

KW - well-covered graph

UR - http://www.scopus.com/inward/record.url?scp=85026768586&partnerID=8YFLogxK

U2 - 10.1016/j.endm.2017.07.038

DO - 10.1016/j.endm.2017.07.038

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AN - SCOPUS:85026768586

SN - 1571-0653

VL - 61

SP - 797

EP - 803

JO - Electronic Notes in Discrete Mathematics

JF - Electronic Notes in Discrete Mathematics

ER -