TY - JOUR

T1 - 1-well-covered graphs revisited

AU - Levit, Vadim E.

AU - Mandrescu, Eugen

N1 - Publisher Copyright:
© 2018 Elsevier Ltd

PY - 2019/8

Y1 - 2019/8

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 any one 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 independent set. Specifically, for well-covered graphs, it means that the vertex v is shedding if and only if G−v is well-covered. In addition, we provide new characterizations of 1-well-covered graphs.

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 any one 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 independent set. Specifically, for well-covered graphs, it means that the vertex v is shedding if and only if G−v is well-covered. In addition, we provide new characterizations of 1-well-covered graphs.

UR - http://www.scopus.com/inward/record.url?scp=85044003619&partnerID=8YFLogxK

U2 - 10.1016/j.ejc.2018.02.021

DO - 10.1016/j.ejc.2018.02.021

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AN - SCOPUS:85044003619

SN - 0195-6698

VL - 80

SP - 261

EP - 272

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

ER -