On the Structure of alpha\ alpha alpha-Stable Graphs (original) (raw)
Related papers
On the structure of -stable graphs
Discrete Mathematics, 2001
The stability number (G) of a graph G is the cardinality of a stability system of G (that is of a stable set of maximum size). A graph is-stable if its stability number remains the same upon both the deletion and the addition of any edge. Trying to generalize some stable trees properties, we show that there does not exist any-stable chordal graph, and we prove that: if G is a connected bipartite graph, then the following assertions are equivalent: (i) G is-stable; (ii) G can be written as a vertex disjoint union of connected bipartite graphs, each of them having exactly two stability systems covering its vertex set; (iii) G has perfect matchings and {M : M is a perfect matching of G} = ∅; (iv) for any vertex of G there are at least two edges incident to this vertex and contained in some perfect matchings; (v) any vertex of G belongs to a cycle, whose edges are alternately in and not in a perfect matching of G; and (vi) {S: S is a stability system of G} = ∅ = {M : M is a maximum matching of G}.
On $ alpha ^{++}$-Stable Graphs
Eprint Arxiv Math 0003057, 2000
The stability number of the graph G, denoted by α(G), is the cardinality of a maximum stable set of G. A graph is well-covered if every maximal stable set has the same size. G is a König-Egerváry graph if its order equals α(G) + µ(G), where µ(G) is the cardinality of a maximum matching in G. In this paper we characterize α ++-stable graphs, namely, the graphs whose stability numbers are invariant to adding any two edges from their complements. We show that a König-Egerváry graph is α ++-stable if and only if it has a perfect matching consisting of pendant edges and no four vertices of the graph span a cycle. As a corollary it gives necessary and sufficient conditions for α ++-stability of bipartite graphs and trees. For instance, we prove that a bipartite graph is α ++-stable if and only if it is well-covered and C4-free.
C O ] 3 0 D ec 1 99 9 On α-Square-Stable Graphs
1999
The stability number of a graph G, denoted by α(G), is the cardinality of a maximum stable set, and μ(G) is the cardinality of a maximum matching in G. If α(G) + μ(G) equals its order, then G is a König-Egerváry graph. We call G an α-square-stable graph if α(G) = α(G), where G denotes the second power of G. These graphs were first investigated by Randerath and Wolkmann, [18]. In this paper we obtain several new characterizations of α-square-stable graphs. We also show that G is an α-square-stable König-Egerváry graph if and only if it has a perfect matching consisting of pendant edges. Moreover, we find that well-covered trees are exactly α-square-stable trees. To verify this result we give a new proof of one Ravindra’s theorem describing well-covered trees, [19].
On alpha\alpha alpha-Square-Stable Graphs
1999
The stability number of a graph G, denoted by alpha(G), is the cardinality of a maximum stable set, and mu(G) is the cardinality of a maximum matching in G. If alpha(G) + mu(G) equals its order, then G is a Koenig-Egervary graph. We call G an alpha\alpha alpha-square-stable graph, shortly square-stable, if alpha(G) = alpha(G*G), where G*G denotes the second power of G. These graphs were first investigated by Randerath and Wolkmann. In this paper we obtain several new characterizations of square-stable graphs. We also show that G is an square-stable Koenig-Egervary graph if and only if it has a perfect matching consisting of pendant edges. Moreover, we find that well-covered trees are exactly square-stable trees. To verify this result we give a new proof of one Ravindra's theorem describing well-covered trees.
On [alpha]+-stable König-Egerváry graphs
Discrete Mathematics, 2003
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 a K onig-Egervà ary graph if its order equals (G) + (G), where (G) is the size of a maximum matching in G. In this paper, we characterize + -stable K onig-Egervà ary graphs, generalizing some previously known results on bipartite graphs and trees. Namely, we prove that a K onig-Egervà ary graph G = (V; 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-Egervà ary graph G to have a perfect matching. Finally, we describe the two following types of + -stable K onig-Egervà ary graphs: those with | {S: S ∈ (G)}| = 0 and | {S: S ∈ (G)}| = 1, respectively.
Arxiv preprint math/9912234, 1999
Abstract: The stability number of a graph G, denoted by alpha (G), is the cardinality of a maximum stable set, and mu (G) is the cardinality of a maximum matching in G. If alpha (G)+ mu (G) equals its order, then G is a Koenig-Egervary graph. We call G an alpha\ alpha alpha-square-stable graph, shortly square-stable, if alpha (G)= alpha (G* G), where G* G denotes the second power of G. These graphs were first investigated by Randerath and Wolkmann. In this paper we obtain several new characterizations of square-stable graphs. We also ...
On α +-stable König–Egerváry graphs
Discrete Mathematics, 2003
The stability number of a graph G, denoted by α(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 α+-stable. G is a König–Egerváry graph if its order equals α(G)+μ(G), where μ(G) is the size of a maximum matching in G. In this paper, we characterize α+-stable König–Egerváry graphs, generalizing some previously known results on bipartite graphs and trees. Namely, we prove that a König–Egerváry graph G=(V,E) of order at least two is α+-stable if and only if G has a perfect matching and (where Ω(G) denotes the family of all maximum stable sets of G). We also show that the equality is a necessary and sufficient condition for a König–Egerváry graph G to have a perfect matching. Finally, we describe the two following types of α+-stable König–Egerváry graphs: those with and , respectively.
C O ] 2 D ec 1 99 9 On α +-Stable König-Egervary Graphs
1999
The stability number of a graph G, denoted by α(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 α-stable. G is a König-Egervary graph if its order equals α(G) + μ(G), where μ(G) is the cardinality of a maximum matching in G. In this paper we characterize α-stable König-Egervary graphs, generalizing some previously known results on bipartite graphs and trees. Namely, we prove that a König-Egervary graph G = (V, E) is α-stable if and only if either |∩{V − S : S ∈ Ω(G)}| = 0, or |∩{V − S : S ∈ Ω(G)}| = 1, and G has a perfect matching (where Ω(G) denotes the family of all maximum stable sets of G). Using this characterization we obtain several new findings on general König-Egervary graphs, for example, the equality |∩{S : S ∈ Ω(G)}| = |∩{V − S : S ∈ Ω(G)}| is a necessary and sufficient condition for a König-Egervary graph G to have a perfect matching.
Discrete Mathematics, 2000
A set I of vertices of a graph G is k-independent if the distance between every two vertices of I is at least k + 1. The k-independence number, k (G), is the cardinality of a maximum k-independent set of G. A set D of vertices of G is k-dominating if every vertex in V (G) − D is at distance at most k from some vertex in D. The k-domination number, k (G), is the cardinality of a minimum k-dominating set of G. A graph G is k-stable (k-stable) if k (G − e) = k (G) (k (G − e) = k (G)) for every edge e of G. We establish conditions under which a graph is k-and k-stable. In particular, we give constructive characterizations of k-and k-stable trees.
On alpha+\ alpha^{+} alpha+-Stable Koenig-Egervary Graphs
Arxiv preprint math/9912022, 1999
The stability number of a graph G, denoted by α(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 α +-stable. G is a König-Egervary graph if its order equals α(G) + µ(G), where µ(G) is the cardinality of a maximum matching in G. In this paper we characterize α +-stable König-Egervary graphs, generalizing some previously known results on bipartite graphs and trees. Namely, we prove that a König-Egervary graph G = (V, E) is α +-stable if and only if either |∩{V − S : S ∈ Ω(G)}| = 0, or |∩{V − S : S ∈ Ω(G)}| = 1, and G has a perfect matching (where Ω(G) denotes the family of all maximum stable sets of G). Using this characterization we obtain several new findings on general König-Egervary graphs, for example, the equality |∩{S : S ∈ Ω(G)}| = |∩{V − S : S ∈ Ω(G)}| is a necessary and sufficient condition for a König-Egervary graph G to have a perfect matching.