TY - JOUR

T1 - Recognizing W2 Graphs

AU - Levit, Vadim E.

AU - Tankus, David

N1 - Publisher Copyright:
© The Author(s) 2024.

PY - 2024/8

Y1 - 2024/8

N2 - Let G be a graph. A set S⊆V(G) is independent if its elements are pairwise nonadjacent. A vertex v∈V(G) is shedding if for every independent set S⊆V(G)\N[v] there exists u∈N(v) such that S∪{u} is independent. An independent set S is maximal if it is not contained in another independent set. An independent set S is maximum if the size of every independent set of G is not bigger than |S|. The size of a maximum independent set of G is denoted α(G). A graph G is well-covered if all its maximal independent sets are maximum, i.e. the size of every maximal independent set is α(G). The graph G belongs to class W2 if every two disjoint independent sets in G are included in two disjoint maximum independent sets. If a graph belongs to the class W2, then it is well-covered. Finding a maximum independent set in an input graph is a well-known NP-hard problem. Recognizing well-covered graphs is co-NP-complete. Recently, it was proved that deciding whether an input graph belongs to the class W2 is co-NP-complete. However, when the input is restricted to well-covered graphs, the complexity status of recognizing graphs in W2 is still not known. In this article, we investigate the connection between shedding vertices and W2 graphs. On the one hand, we prove that recognizing shedding vertices is co-NP-complete. On the other hand, we find polynomial solutions for restricted cases of the problem. We also supply polynomial characterizations of several families of W2 graphs.

AB - Let G be a graph. A set S⊆V(G) is independent if its elements are pairwise nonadjacent. A vertex v∈V(G) is shedding if for every independent set S⊆V(G)\N[v] there exists u∈N(v) such that S∪{u} is independent. An independent set S is maximal if it is not contained in another independent set. An independent set S is maximum if the size of every independent set of G is not bigger than |S|. The size of a maximum independent set of G is denoted α(G). A graph G is well-covered if all its maximal independent sets are maximum, i.e. the size of every maximal independent set is α(G). The graph G belongs to class W2 if every two disjoint independent sets in G are included in two disjoint maximum independent sets. If a graph belongs to the class W2, then it is well-covered. Finding a maximum independent set in an input graph is a well-known NP-hard problem. Recognizing well-covered graphs is co-NP-complete. Recently, it was proved that deciding whether an input graph belongs to the class W2 is co-NP-complete. However, when the input is restricted to well-covered graphs, the complexity status of recognizing graphs in W2 is still not known. In this article, we investigate the connection between shedding vertices and W2 graphs. On the one hand, we prove that recognizing shedding vertices is co-NP-complete. On the other hand, we find polynomial solutions for restricted cases of the problem. We also supply polynomial characterizations of several families of W2 graphs.

UR - http://www.scopus.com/inward/record.url?scp=85189186682&partnerID=8YFLogxK

U2 - 10.1007/s00373-024-02777-6

DO - 10.1007/s00373-024-02777-6

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AN - SCOPUS:85189186682

SN - 0911-0119

VL - 40

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

IS - 4

M1 - 72

ER -