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C. Balbuena 《Discrete Mathematics》2008,308(16):3526-3536
For a connected graph G, the rth extraconnectivity κr(G) is defined as the minimum cardinality of a cutset X such that all remaining components after the deletion of the vertices of X have at least r+1 vertices. The standard connectivity and superconnectivity correspond to κ0(G) and κ1(G), respectively. The minimum r-tree degree of G, denoted by ξr(G), is the minimum cardinality of N(T) taken over all trees T⊆G of order |V(T)|=r+1, N(T) being the set of vertices not in T that are neighbors of some vertex of T. When r=1, any such considered tree is just an edge of G. Then, ξ1(G) is equal to the so-called minimum edge-degree of G, defined as ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}, where d(u) stands for the degree of vertex u. A graph G is said to be optimally r-extraconnected, for short κr-optimal, if κr(G)?ξr(G). In this paper, we present some sufficient conditions that guarantee κr(G)?ξr(G) for r?2. These results improve some previous related ones, and can be seen as a complement of some others which were obtained by the authors for r=1. 相似文献
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C. Balbuena 《Discrete Mathematics》2007,307(6):659-667
The restricted connectivity κ′(G) of a connected graph G is defined as the minimum cardinality of a vertex-cut over all vertex-cuts X such that no vertex u has all its neighbors in X; the superconnectivity κ1(G) is defined similarly, this time considering only vertices u in G-X, hence κ1(G)?κ′(G). The minimum edge-degree of G is ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}, d(u) standing for the degree of a vertex u. In this paper, several sufficient conditions yielding κ1(G)?ξ(G) are given, improving a previous related result by Fiol et al. [Short paths and connectivity in graphs and digraphs, Ars Combin. 29B (1990) 17-31] and guaranteeing κ1(G)=κ′(G)=ξ(G) under some additional constraints. 相似文献
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C. Balbuena 《Discrete Mathematics》2008,308(10):1985-1993
A matched sum graph G of two graphs G1 and G2 of the same order is obtained from the union of G1 and G2 and from joining each vertex of G1 with one vertex of G2 according to one bijection f between the vertices in V(G1) and V(G2). When G1=G2=H then f is just a permutation of V(H) and the corresponding matched sum graph is a permutation graph Hf. In this paper, we derive lower bounds for the connectivity, edge-connectivity, and different conditional connectivities in matched sum graphs, and present sufficient conditions which guarantee maximum values for these conditional connectivities. 相似文献
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A graph is superconnected, for short super-κ, if all minimum vertex-cuts consist of the vertices adjacent with one vertex. In this paper we prove for any r-regular graph of diameter D and odd girth g that if D≤g−2, then the graph is super-κ when g≥5 and a complete graph otherwise. 相似文献
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