Abstract A set S ⊆ V ( G ) is stable (or independent) if no two vertices from S are adjacent. Let... more Abstract A set S ⊆ V ( G ) is stable (or independent) if no two vertices from S are adjacent. Let Ψ ( G ) be the family of all local maximum stable sets [V. E. Levit, E. Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discr. Appl. Math. 124 (2002) 91–101] of graph G, i.e., S ∈ Ψ ( G ) if S is a maximum stable set of the subgraph induced by S ∪ N ( S ) , where N(S) is the neighborhood of S. If I is stable and there is a matching from N(I) into I, then I is a crown of order | I | + | N ( I ) | , and we write I ∈ Crown ( G ) [F. N. Abu-Khzam, M. R. Fellows, M. A. Langston, W. H. Suters, Crown structures for vertex cover kernelization, Theory of Comput. Syst. 41 (2007) 411–430]. In this paper we show that Crown ( G ) ⊆ Ψ ( G ) holds for every graph, while Crown ( G ) = Ψ ( G ) is true for bipartite and very well-covered graphs. For general graphs, it is NP-complete to decide if a graph has a crown of a given order [C. Sloper. Techniques in Parameterized Algorithm Design. Ph.D. thesis, Department of Computer Science, University of Bergen, Norway, 2005]. We prove that in a bipartite graph G with a unique perfect matching, there exist crowns of every possible even order.
The stability number of a graph G, denoted by � (G), is the cardinality of a stable set of maximu... more The stability number of a graph G, denoted by � (G), is the cardinality of a stable set of maximum size in G. If its stability number remains the same upon the addition of any edge, then G is called � + -stable. G is aK # onig–Egerva E) of order at least two is � + -stable if and only if G has a perfect matching and | {V − S: S ∈ � (G)}| 6 1 (where � (G) denotes the family of all maximum stable sets of G). We also show that the equality | {V − S: S ∈ � (G)}| = | {S: S ∈ � (G)}| is a necessary and su>cient condition for a K# onig–Egerv&ary graph G to have a perfect matching. Finally, we describe the two following types of � + -stable K#onig–Egerv&ary graphs: those with | {S: S ∈ � (G)}| = 0 and | {S: S ∈ � (G)}| = 1, respectively. c
Electronic Notes in Discrete Mathematics, Oct 1, 2016
Abstract A set S ⊆ V ( G ) is stable (or independent) if no two vertices from S are adjacent. Let... more Abstract A set S ⊆ V ( G ) is stable (or independent) if no two vertices from S are adjacent. Let Ψ ( G ) be the family of all local maximum stable sets [V. E. Levit, E. Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discr. Appl. Math. 124 (2002) 91–101] of graph G, i.e., S ∈ Ψ ( G ) if S is a maximum stable set of the subgraph induced by S ∪ N ( S ) , where N(S) is the neighborhood of S. If I is stable and there is a matching from N(I) into I, then I is a crown of order | I | + | N ( I ) | , and we write I ∈ Crown ( G ) [F. N. Abu-Khzam, M. R. Fellows, M. A. Langston, W. H. Suters, Crown structures for vertex cover kernelization, Theory of Comput. Syst. 41 (2007) 411–430]. In this paper we show that Crown ( G ) ⊆ Ψ ( G ) holds for every graph, while Crown ( G ) = Ψ ( G ) is true for bipartite and very well-covered graphs. For general graphs, it is NP-complete to decide if a graph has a crown of a given order [C. Sloper. Techniques in Parameterized Algorithm Design. Ph.D. thesis, Department of Computer Science, University of Bergen, Norway, 2005]. We prove that in a bipartite graph G with a unique perfect matching, there exist crowns of every possible even order.
In this paper we show that for any integer $alpha$ > 7, there exists a (dis)connected well-covere... more In this paper we show that for any integer $alpha$ > 7, there exists a (dis)connected well-covered graph $G$ with $alpha$ = alpha(G), whose independence polynomial is not unimodal. In addition, we present a number of sufficient conditions for a graph $G$ with alpha(G) < 7 to have unimodal independence polynomial.
Abstract A set S ⊆ V ( G ) is stable (or independent) if no two vertices from S are adjacent. Let... more Abstract A set S ⊆ V ( G ) is stable (or independent) if no two vertices from S are adjacent. Let Ψ ( G ) be the family of all local maximum stable sets [V. E. Levit, E. Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discr. Appl. Math. 124 (2002) 91–101] of graph G, i.e., S ∈ Ψ ( G ) if S is a maximum stable set of the subgraph induced by S ∪ N ( S ) , where N(S) is the neighborhood of S. If I is stable and there is a matching from N(I) into I, then I is a crown of order | I | + | N ( I ) | , and we write I ∈ Crown ( G ) [F. N. Abu-Khzam, M. R. Fellows, M. A. Langston, W. H. Suters, Crown structures for vertex cover kernelization, Theory of Comput. Syst. 41 (2007) 411–430]. In this paper we show that Crown ( G ) ⊆ Ψ ( G ) holds for every graph, while Crown ( G ) = Ψ ( G ) is true for bipartite and very well-covered graphs. For general graphs, it is NP-complete to decide if a graph has a crown of a given order [C. Sloper. Techniques in Parameterized Algorithm Design. Ph.D. thesis, Department of Computer Science, University of Bergen, Norway, 2005]. We prove that in a bipartite graph G with a unique perfect matching, there exist crowns of every possible even order.
The stability number of a graph G, denoted by � (G), is the cardinality of a stable set of maximu... more The stability number of a graph G, denoted by � (G), is the cardinality of a stable set of maximum size in G. If its stability number remains the same upon the addition of any edge, then G is called � + -stable. G is aK # onig–Egerva E) of order at least two is � + -stable if and only if G has a perfect matching and | {V − S: S ∈ � (G)}| 6 1 (where � (G) denotes the family of all maximum stable sets of G). We also show that the equality | {V − S: S ∈ � (G)}| = | {S: S ∈ � (G)}| is a necessary and su>cient condition for a K# onig–Egerv&ary graph G to have a perfect matching. Finally, we describe the two following types of � + -stable K#onig–Egerv&ary graphs: those with | {S: S ∈ � (G)}| = 0 and | {S: S ∈ � (G)}| = 1, respectively. c
Electronic Notes in Discrete Mathematics, Oct 1, 2016
Abstract A set S ⊆ V ( G ) is stable (or independent) if no two vertices from S are adjacent. Let... more Abstract A set S ⊆ V ( G ) is stable (or independent) if no two vertices from S are adjacent. Let Ψ ( G ) be the family of all local maximum stable sets [V. E. Levit, E. Mandrescu, A new greedoid: the family of local maximum stable sets of a forest, Discr. Appl. Math. 124 (2002) 91–101] of graph G, i.e., S ∈ Ψ ( G ) if S is a maximum stable set of the subgraph induced by S ∪ N ( S ) , where N(S) is the neighborhood of S. If I is stable and there is a matching from N(I) into I, then I is a crown of order | I | + | N ( I ) | , and we write I ∈ Crown ( G ) [F. N. Abu-Khzam, M. R. Fellows, M. A. Langston, W. H. Suters, Crown structures for vertex cover kernelization, Theory of Comput. Syst. 41 (2007) 411–430]. In this paper we show that Crown ( G ) ⊆ Ψ ( G ) holds for every graph, while Crown ( G ) = Ψ ( G ) is true for bipartite and very well-covered graphs. For general graphs, it is NP-complete to decide if a graph has a crown of a given order [C. Sloper. Techniques in Parameterized Algorithm Design. Ph.D. thesis, Department of Computer Science, University of Bergen, Norway, 2005]. We prove that in a bipartite graph G with a unique perfect matching, there exist crowns of every possible even order.
In this paper we show that for any integer $alpha$ > 7, there exists a (dis)connected well-covere... more In this paper we show that for any integer $alpha$ > 7, there exists a (dis)connected well-covered graph $G$ with $alpha$ = alpha(G), whose independence polynomial is not unimodal. In addition, we present a number of sufficient conditions for a graph $G$ with alpha(G) < 7 to have unimodal independence polynomial.
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