K-Independence Stable Graphs Upon Edge Removal (original) (raw)
Related papers
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 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 the Structure of alpha\ alpha alpha-Stable Graphs
Arxiv preprint math/9911227, 1999
The stability number α(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 α-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 α-stable chordal graph, and we prove that: if G is a connected bipartite graph, then the following assertions are equivalent: G is α-stable; 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; G has perfect matchings and ∩{M : M is a perfect matching of G} = ∅; from each vertex of G are issuing at least two edges, contained in some perfect matchings of G; any vertex of G belongs to a cycle, whose edges are alternately in and not in a perfect matching of G; ∩{S : S is a stability system of G} = ∅ = ∩{M : M is a maximum matching of G}.
Introduction to kkk-independent graph of a graph
2015
Let G=(V,E)G=(V,E)G=(V,E) be a simple graph. A set IsubseteqVI\subseteq VIsubseteqV is an independent set, if no two of its members are adjacent in GGG. The kkk-independent graph of GGG, Ik(G)I_k (G)Ik(G), is defined to be the graph whose vertices correspond to the independent sets of GGG that have cardinality at most kkk. Two vertices in Ik(G)I_k(G)Ik(G) are adjacent if and only if the corresponding independent sets of GGG differ by either adding or deleting a single vertex. In this paper, we obtain some properties of Ik(G)I_k(G)Ik(G) and compute it for some graphs.
On the number of vertices belonging to all maximum stable sets of a graph
Discrete Applied Mathematics, 2002
Let us denote by α(G) the size of a maximum stable set, and by μ(G) the size of a maximum matching of a graph G, and let ξ(G) be the number of vertices which belong to all maximum stable sets. We shall show that ξ(G)⩾1+α(G)−μ(G) holds for any connected graph, whenever α(G)>μ(G). This inequality improves on related results by Hammer et al. (SIAM J. Algebraic Discrete Methods 3 (1982) 511) and by Levit and Mandrescu [(prE-print math. CO/9912047 (1999) 13pp.)].We also prove that on one hand, ξ(G)>0 can be recognized in polynomial time whenever μ(G)<|V(G)|/3, and on the other hand determining whether ξ(G)>k is, in general, NP-complete for any fixed k⩾0.
THE k-INDEPENDENT GRAPH OF A GRAPH
Advances and Applications in Discrete Mathematics, 2017
Let G = (V, E) be a simple graph. A set I ⊆ V is an independent set, if no two of its members are adjacent in G. The k-independent graph of G, I k (G), is defined to be the graph whose vertices correspond to the independent sets of G that have cardinality at most k. Two vertices in I k (G) are adjacent if and only if the corresponding independent sets of G differ by either adding or deleting a single vertex. In this paper, we obtain some properties of I k (G) and compute it for some graphs.
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].
K-Independence Critical Graphs
Australas. J Comb., 2012
Let k be a positive integer and G = (V (G), E(G)) a graph. A subset S of V (G) is a k-independent set of G if the subgraph induced by the vertices of S has maximum degree at most k − 1. The maximum cardinality of a k-independent set of G is the k-independence number β k (G). In this paper, we study the properties of graphs for which the k-independence number changes whenever an edge or vertex is removed or an edge is added.
Maximal k-independent sets in graphs
Discussiones Mathematicae Graph Theory, 2008
A subset of vertices of a graph G is k-independent if it induces in G a subgraph of maximum degree less than k. The minimum and maximum cardinalities of a maximal k-independent set are respectively denoted i k (G) and β k (G). We give some relations between β k (G) and β j (G) and between i k (G) and i j (G) for j = k. We study two families of extremal graphs for the inequality i 2 (G) ≤ i(G) + β(G). Finally we give an upper bound on i 2 (G) and a lower bound when G is a cactus.
The intersection of all maximum stable sets of a tree and its pendant vertices
Discrete Mathematics, 2008
A stable set in a graph G is a set of mutually non-adjacent vertices, α(G) is the size of a maximum stable set of G, and core(G) is the intersection of all its maximum stable sets. In this paper we demonstrate that in a tree T , of order n ≥ 2, any stable set of size ≥ n/2 contains at least one pendant vertex. Hence, we deduce that any maximum stable set in a tree contains at least one pendant vertex. We give a new proof for a theorem of Hopkins and Staton [5] characterizing strong unique trees. Using this result we show that if {A, B} is the bipartition of a tree T and S is a stable set with |S| > min{|A| , |B|}, then S contains at least a pendant vertex.