Year: 2025
Type: article
Abstract: A graph G is well-covered if all its maximal independent sets are of the same cardinality. Let $$w:V(G) \longrightarrow \mathbb {R}$$ w : V ( G ) ⟶ R be a weight function. Then G is w-well-covered if all its maximal independent sets are of the same weight. An edge $$xy \in E(G)$$ x y ∈ E ( G ) is relating if there exists $$S \subseteq V(G)$$ S ⊆ V ( G ) such that both $$S \cup \{x\}$$ S ∪ { x } and $$S \cup \{y\}$$ S ∪ { y } are maximal independent sets. If xy is relating then $$ w(x)=w(y)$$ w ( x ) = w ( y ) for every weight function w such that G is w-well-covered. Relating edges are of crucial importance for investigating w-well-covered graphs. The problem whether an edge is relating is NP-complete. We prove that this problem remains NP-complete even for graphs without cycles of length 6. A graph G belongs to the class $$\mathbf {W_{2}}$$ W 2 if every two pairwise disjoint independent sets in G are included in two pairwise disjoint maximum independent sets. A vertex $$v \in V(G)$$ v ∈ V ( G ) is shedding if for every independent set $$S \subseteq V(G) \setminus N[v]$$ S ⊆ V ( G ) \ N [ v ] there exists $$u \in N(v)$$ u ∈ N ( v ) such that $$S \cup \{u\}$$ S ∪ { u } is independent. Shedding vertices play an important role in studying the class $$\mathbf {W_{2}}$$ W 2 . Recognizing shedding vertices is co-NP-complete. We prove that this problem is co-NP-complete even for graphs without cycles of length 6.
Source: Graphs and Combinatorics
Authors: Vadim E. Levit, David Tankus
Institutions: Ariel University, Sami Shamoon College of Engineering
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,190
Funder: Ariel University
OpenAlex ID: W4410466516