Given a graph G and a non-negative integer h, the h-restricted connectivity κh（G） of G is the minimum cardinality of a set of vertices of G, in which at least h neighbors of any vertex is not included, if any, whos...Given a graph G and a non-negative integer h, the h-restricted connectivity κh（G） of G is the minimum cardinality of a set of vertices of G, in which at least h neighbors of any vertex is not included, if any, whose deletion disconnects G and every remaining component has the minimum degree of vertex at least h; and the h-extra connectivity κh（G） of G is the minimum cardinality of a set of vertices of G, if any, whose deletion disconnects G and every remaining component has order more than h. This paper shows that for the hypercube Qn and the folded hypercube FQn, κ1（Qn）=κ（1）（Qn）=2n-2 for n≥3, κ2（Qn）=3n-5 for n≥4, κ1（FQn）=κ（1）（FQn）=2n for n≥4 and κ（2）（FQn）=4n-4 for n≥8.展开更多
An m-restricted edge cut is an edge cut that separates a connected graph into a disconnected one with no components having order less than m. m-restricted edge connectivity λ<SUB> m </SUB>is the c...An m-restricted edge cut is an edge cut that separates a connected graph into a disconnected one with no components having order less than m. m-restricted edge connectivity λ<SUB> m </SUB>is the cardinality of a minimum m-restricted edge cut. Let G be a connected k-regular graph of order at least 2m that contains m-restricted edge cuts and X be a subgraph of G. Let ∂(X) denote the number of edges with one end in X and the other not in X and ξ<SUB> m </SUB>= min{∂(X) : X is a connected vertex-induced subgraph of order m}. It is proved in this paper that if G has girth at least m/2+ 2, then λ<SUB> m </SUB>≤ ξ<SUB> m </SUB>. The upper bound of λ<SUB> m </SUB>is sharp.展开更多
Abstract This paper studies restricted fault diameter of the n-dimensional hypercube networks Qn (n S 2). It is shown that for arbitrary two vertices x and y with the distance d in Qn and any set F with at most 2nm3 v...Abstract This paper studies restricted fault diameter of the n-dimensional hypercube networks Qn (n S 2). It is shown that for arbitrary two vertices x and y with the distance d in Qn and any set F with at most 2nm3 vertices in Qn m {x,y}, if F contains neither of neighbor-sets of x and y in Qn, then the distance between x and y in Qn m F is vigen byFurthermore, the upper bounds are tight. As an immediately consequence, Qn can tolerate up to 2nm3 vertices failures and remain diameter 4 if n=3 and n+2 if nS4 provided that for each vertex x in Qn, all the neighbors of x do not fail at the same time. This improves Esfahanian's result.展开更多
Restricted edge connectivity of a graph G is defined to be the minimum size |U| of a set U of edges such that G-U is disconnected and G-U contains no trivial component K1. The high order edge connectivity Ni, i1, is t...Restricted edge connectivity of a graph G is defined to be the minimum size |U| of a set U of edges such that G-U is disconnected and G-U contains no trivial component K1. The high order edge connectivity Ni, i1, is the number of edge outsets of size i. TO determine all Ni, i 1, for a general graph is NP-hard. In this paper, the authors evaluated the restricted edge connectivity and the high order edge connectivity Ni, 1 i -1, for any connected Abelian Cayley graphs explicitly.展开更多
文摘Given a graph G and a non-negative integer h, the h-restricted connectivity κh（G） of G is the minimum cardinality of a set of vertices of G, in which at least h neighbors of any vertex is not included, if any, whose deletion disconnects G and every remaining component has the minimum degree of vertex at least h; and the h-extra connectivity κh（G） of G is the minimum cardinality of a set of vertices of G, if any, whose deletion disconnects G and every remaining component has order more than h. This paper shows that for the hypercube Qn and the folded hypercube FQn, κ1（Qn）=κ（1）（Qn）=2n-2 for n≥3, κ2（Qn）=3n-5 for n≥4, κ1（FQn）=κ（1）（FQn）=2n for n≥4 and κ（2）（FQn）=4n-4 for n≥8.
基金National Natural Science Foundation of China (Grant No.10271105) and Doctoral Fund of Zhangzhou Normal College.
文摘An m-restricted edge cut is an edge cut that separates a connected graph into a disconnected one with no components having order less than m. m-restricted edge connectivity λ<SUB> m </SUB>is the cardinality of a minimum m-restricted edge cut. Let G be a connected k-regular graph of order at least 2m that contains m-restricted edge cuts and X be a subgraph of G. Let ∂(X) denote the number of edges with one end in X and the other not in X and ξ<SUB> m </SUB>= min{∂(X) : X is a connected vertex-induced subgraph of order m}. It is proved in this paper that if G has girth at least m/2+ 2, then λ<SUB> m </SUB>≤ ξ<SUB> m </SUB>. The upper bound of λ<SUB> m </SUB>is sharp.
基金Supported by ANSF (No.01046102) and NNSF of China (No.10271114).
文摘Abstract This paper studies restricted fault diameter of the n-dimensional hypercube networks Qn (n S 2). It is shown that for arbitrary two vertices x and y with the distance d in Qn and any set F with at most 2nm3 vertices in Qn m {x,y}, if F contains neither of neighbor-sets of x and y in Qn, then the distance between x and y in Qn m F is vigen byFurthermore, the upper bounds are tight. As an immediately consequence, Qn can tolerate up to 2nm3 vertices failures and remain diameter 4 if n=3 and n+2 if nS4 provided that for each vertex x in Qn, all the neighbors of x do not fail at the same time. This improves Esfahanian's result.
文摘Restricted edge connectivity of a graph G is defined to be the minimum size |U| of a set U of edges such that G-U is disconnected and G-U contains no trivial component K1. The high order edge connectivity Ni, i1, is the number of edge outsets of size i. TO determine all Ni, i 1, for a general graph is NP-hard. In this paper, the authors evaluated the restricted edge connectivity and the high order edge connectivity Ni, 1 i -1, for any connected Abelian Cayley graphs explicitly.
