TY - JOUR

T1 - Well-covered graphs without cycles of lengths 4, 5 and 6

AU - Levit, Vadim E.

AU - Tankus, David

N1 - Publisher Copyright:
© 2015 Elsevier B.V. All rights reserved.

PY - 2015

Y1 - 2015

N2 - A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight, where a weight of a set is the sum of weights of its elements. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input graph G without cycles of lengths 4, 5, and 6, we characterize polynomially the vector space of weight functions w for which G is w-well-covered. Let B be an induced complete bipartite subgraph of G on vertex sets of bipartition BX and BY. Assume that there exists an independent set S such that each of S ∪ BX and S ∪ BY is a maximal independent set of G. Then B is a generating subgraph of G, and it produces the restriction w(BX) = w(BY). It is easy to see that for every weight function w, if G is w-well-covered, then the above restriction is satisfied. In the special case, where BX = {x} and BY = {y}, we say that xy is a relating edge. Recognizing relating edges and generating subgraphs is an NP-complete problem. However, we provide a polynomial algorithm for recognizing generating subgraphs of an input graph without cycles of lengths 5, 6 and 7. We also present a polynomial algorithm for recognizing relating edges in an input graph without cycles of lengths 5 and 6.

AB - A graph G is well-covered if all its maximal independent sets are of the same cardinality. Assume that a weight function w is defined on its vertices. Then G is w-well-covered if all maximal independent sets are of the same weight, where a weight of a set is the sum of weights of its elements. For every graph G, the set of weight functions w such that G is w-well-covered is a vector space. Given an input graph G without cycles of lengths 4, 5, and 6, we characterize polynomially the vector space of weight functions w for which G is w-well-covered. Let B be an induced complete bipartite subgraph of G on vertex sets of bipartition BX and BY. Assume that there exists an independent set S such that each of S ∪ BX and S ∪ BY is a maximal independent set of G. Then B is a generating subgraph of G, and it produces the restriction w(BX) = w(BY). It is easy to see that for every weight function w, if G is w-well-covered, then the above restriction is satisfied. In the special case, where BX = {x} and BY = {y}, we say that xy is a relating edge. Recognizing relating edges and generating subgraphs is an NP-complete problem. However, we provide a polynomial algorithm for recognizing generating subgraphs of an input graph without cycles of lengths 5, 6 and 7. We also present a polynomial algorithm for recognizing relating edges in an input graph without cycles of lengths 5 and 6.

KW - Generating subgraph

KW - Independent set

KW - Relating edge

KW - Well-covered graph

UR - http://www.scopus.com/inward/record.url?scp=84933280238&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2015.01.001

DO - 10.1016/j.dam.2015.01.001

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AN - SCOPUS:84933280238

SN - 0166-218X

VL - 186

SP - 158

EP - 167

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

IS - 1

ER -