Minimum K-restricted Edge Cut Research Articles

Sufficient conditions for graphs with girth [formula omitted] to be maximally [formula omitted]-restricted edge connected

For a connected graph G=(V,E), an edge set S⊆E is a k-restricted edge cut if G−S is disconnected and every component of G−S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. Let ξk(G)=min{|[X,X̄]|:|X|=k,G[X]is connected}, where X̄=V∖X. G is maximally k-restricted edge connected (λk-optimal for short) if λk(G)=ξk(G). The k-restricted edge connectivity is more refined network reliability index than the edge connectivity. In this paper, for graph G with girth g≥2k+1 and minimum degree δ≥k, we present a lower bound on the cardinality of λk-atoms of G when λk(G)<ξk(G). Moreover, we give some sufficient conditions for graphs with girth g to be λk-optimal.

On computing of a conditional edge connectivity of alternating group network

An edge subset F of a connected graph G is called a k-restricted edge cut if is disconnected and every vertex in has at least k neighbours in . The cardinality of a minimum k-restricted edge-cut of G is called the k-restricted edge connectivity of G, denoted by . The alternating group network () is defined as the Cayley graph on the alternating group of degree n with respect to . In this paper, we prove that for each .

Sufficient conditions for triangle-free graphs to be super k-restricted edge-connected

An edge cut S of a connected graph G=(V,E) is a k-restricted edge cut if every component of G−S contains at least k vertices. A graph is said to be super k-restricted edge-connected if every minimum k-restricted edge cut is a set of edges incident to a certain connected subgraph of order k. Let k be a positive integer, and let G be a connected triangle-free graph of order n≥2k. In this paper, we prove that if the minimum degree δ(G)≥k+1−(−1)k and there are at least k+1+(−1)k2 common vertices in the neighbor sets of each pair of nonadjacent vertices in G, then G is super k-restricted edge-connected.

Degree conditions for graphs to be maximally [formula omitted]-restricted edge connected and super [formula omitted]-restricted edge connected

For a connected graph G=(V,E), an edge set S⊆E(G) is called a k-restricted edge cut of G if G−S is disconnected and every component of G−S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. For two disjoint vertex subsets X,Y of G, define [X,Y]={xy∈E(G):x∈X,y∈Y} and define ξk(G)=min{|[X,X¯]|:X⊆V(G),|X|=k,G[X]is connected}, where X¯=V(G)∖X. G is λk-optimal if λk(G)=ξk(G). Furthermore, G is super-λk if every minimum k-restricted edge cut of G isolates a connected subgraph with order k. The k-restricted edge connectivity is an important index to estimate the reliability of networks. In this paper, some degree conditions for graphs to be maximally k-restricted edge connected and super k-restricted edge connected are given.

Formula omitted]-restricted edge connectivity in [formula omitted]-clique-free graphs

Let G be a graph with vertex set V(G) and edge set E(G). An edge subset S⊆E(G) is called a k-restricted edge cut if G−S is not connected and every component of G−S has at least k vertices. The k-restricted edge connectivity of a connected graph G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. Let [X,X̄] denote the set of edges between a vertex set X⊂V(G) and its complement X̄=V(G)∖X. A vertex set X⊂V(G) is called a λk-fragment if [X,X̄] is a minimum k-restricted edge cut of G. Let ξk(G)=min{|[X,X̄]|:|X|=k,G[X]is connected}. In this work, we give a lower bound on the cardinality of λk-fragments of a graph G satisfying λk(G)<ξk(G) and containing no (p+1)-cliques. As a consequence of this result, we show a sufficient condition for a graph G with λk(G)=ξk(G).

Sufficient conditions for k-restricted edge connected graphs

For a connected graph G=(V,E), an edge set S⊆E is a k-restricted edge cut if G−S is disconnected and every component of G−S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. Let ξk(G)=min⁡{|[X,X¯]|:|X|=k,G[X]is connected}, where X¯=V\\X. G is maximally k-restricted edge connected (λk-optimal for short) if λk(G)=ξk(G). The k-restricted edge connectivity is more refined network reliability indices than edge connectivity. In this paper, let k≥2 be an integer, and let G be a graph of order ν(G) at least 2k satisfying |N(u)∩N(v)|≥2k−2 for all pairs u,v of nonadjacent vertices. If for each triangle T there exists at least one vertex v∈V(T) such that d(v)≥⌊ν(G)2⌋+k−1, then G is λk-optimal.

A sufficient condition for graphs to be [formula omitted]-optimal

For a connected graph G=(V,E), an edge set S⊆E is a k-restricted edge cut if G−S is disconnected and every component of G−S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. Let ξk(G)=min{|[X,X̄]|:|X|=k,G[X] is connected}. G is λk-optimal if λk(G)=ξk(G). In 2004, Hellwig and Volkmann gave a sufficient condition for λ2-optimality in graphs of diameter 2. In this paper, we extend the result and give a similar sufficient condition for λk-optimality in graphs of diameter 2 with k≥3.

Optimally restricted edge connected elementary Harary graphs

An edge subset F of a connected graph G=(V,E) is a k-restricted edge cut if G−F is disconnected, and every component of G−F has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is the cardinality of a minimum k-restricted edge cut. By the current studies on λk, it can be seen that the larger λk is, the more reliable the graph is. Hence one expects λk to be as large as possible. A possible upper bound for λk is ξk defined as ξk(G)=min{ω(S):0̸≠S⊂V(G),|S|=k and G[S] is connected }, where ω(S) is the number of edges with one end in S and the other end in V(G)∖S, and G[S] is the subgraph of G induced by S. A graph G is called λk-optimal if λk(G)=ξk(G). A natural question is whether there exists a graph G which is λk-optimal for any k≤|V(G)|/2. In this paper, we show that except for two cases, the elementary Harary graph has this property.

A neighborhood condition for graphs to be maximally k-restricted edge connected

For a connected graph G = ( V , E ) , an edge set S ⊆ E is a k-restricted edge cut if G − S is disconnected and every component of G − S has at least k vertices. The k-restricted edge connectivity of G, denoted by λ k ( G ) , is defined as the cardinality of a minimum k-restricted edge cut. Let ξ k ( G ) = min { | [ X , X ¯ ] | : | X | = k , G [ X ] is connected } . G is λ k -optimal if λ k ( G ) = ξ k ( G ) . Let k ⩾ 4 be an integer. In this paper, we show that if | N G ( u ) ∩ N G ( v ) | ⩾ k for all pairs u , v of non-adjacent vertices and ξ k ( G ) ⩽ ⌊ | V | 2 ⌋ + k , then G is λ k -optimal.

The k-Restricted Edge Connectivity of Balanced Bipartite Graphs

For a connected graph G = (V, E), an edge set $${S\subset E}$$ is called a k-restricted edge cut if G − S is disconnected and every component of G − S contains at least k vertices. The k-restricted edge connectivity of G, denoted by λ k (G), is defined as the cardinality of a minimum k-restricted edge cut. For two disjoint vertex sets $${U_1,U_2\subset V(G)}$$, denote the set of edges of G with one end in U 1 and the other in U 2 by [U 1, U 2]. Define $${\xi_k(G)=\min\{|[U,V(G){\setminus} U]|: U}$$ is a vertex subset of order k of G and the subgraph induced by U is connected}. A graph G is said to be λ k -optimal if λk (G) = ξ k (G). A graph is said to be super-λk if every minimum k-restricted edge cut is a set of edges incident to a certain connected subgraph of order k. In this paper, we present some degree-sum conditions for balanced bipartite graphs to be λk -optimal or super-λk . Moreover, we demonstrate that our results are best possible.

A proof of an inequality concerning k-restricted edge connectivity

An edge subset S of a connected graph G is a k-restricted edge cut if G - S is disconnected, and every component of G - S has at least k vertices. The k-restricted edge connectivity is the cardinality of a minimum k-restricted edge cut. In this note, we show that except for a well-defined class of graphs, k-restricted edge cuts of a connected graph G exist for any k ⩽ δ ( G ) + 1 , where δ ( G ) is the minimum degree of G. Furthermore, we obtain an upper bound for k-restricted edge connectivity.