Restricted Edge Connected Research Articles

Super (Edge-)connectivity of the Folded Petersen Network

Reliability evaluation of an interconnection network is of great significance for construction and maintenance of the network. The super (edge-)connectivity, restricted (edge-)connectivity and cyclic (edge-)connectivity are important parameters to evaluate network reliability. Compared with the hypercube [Formula: see text], the folded Petersen network [Formula: see text] has better properties, including strong connectivity, symmetry. Besides, it has smaller diameter and more vertices than [Formula: see text] with the same degree and connectivity. In this paper, based on the symmetric properties of [Formula: see text], we prove that [Formula: see text] is super (edge-)connected, super restricted edge-connected and super cyclically edge-connected. In addition, we obtain the restricted edge-connectivity and the cyclic edge-connectivity of [Formula: see text].

The Restricted Edge-Connectivity of Strong Product Graphs

The restricted edge-connectivity of a connected graph G, denoted by λ′(G), if it exists, is the minimum cardinality of a set of edges whose deletion makes G disconnected, and each component has at least two vertices. It was proved that λ′(G) exists if and only if G has at least four vertices and G is not a star. In this case, a graph G is called maximally restricted edge-connected if λ′(G)=ξ(G), and a graph G is called super restricted edge-connected if each minimum restricted edge-cut isolates an edge of G. The strong product of graphs G and H, denoted by G⊠H, is the graph with the vertex set V(G)×V(H) and the edge set {(x1,y1)(x2,y2)|x1=x2 and y1y2∈E(H); or y1=y2 and x1x2∈E(G); or x1x2∈E(G) and y1y2∈E(H)}. In this paper, we determine, for any nontrivial connected graph G, the restricted edge-connectivity of G⊠Pn, G⊠Cn and G⊠Kn, where Pn, Cn and Kn are the path, cycle and complete graph of order n, respectively. As corollaries, we give sufficient conditions for these strong product graphs G⊠Pn, G⊠Cn and G⊠Kn to be maximally restricted edge-connected and super restricted edge-connected.

On the Super (Edge)-Connectivity of Generalized Johnson Graphs

Let [Formula: see text], [Formula: see text] and [Formula: see text] be non-negative integers. The generalized Johnson graph [Formula: see text] is the graph whose vertices are the [Formula: see text]-subsets of the set [Formula: see text], and two vertices are adjacent if and only if they intersect with [Formula: see text] elements. Special cases of generalized Johnson graph include the Kneser graph [Formula: see text] and the Johnson graph [Formula: see text]. These graphs play an important role in coding theory, Ramsey theory, combinatorial geometry and hypergraphs theory. In this paper, we discuss the connectivity properties of the Kneser graph [Formula: see text] and [Formula: see text] by their symmetric properties. Specifically, with the help of some properties of vertex/edge-transitive graphs we prove that [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text] are super restricted edge-connected. Besides, we obtain the exact value of the restricted edge-connectivity and the cyclic edge-connectivity of the Kneser graph [Formula: see text] [Formula: see text] and [Formula: see text] [Formula: see text], and further show that the Kneser graph [Formula: see text] [Formula: see text] is super vertex-connected and super cyclically edge-connected.

The Super Restricted Edge-connectedness of Direct Product Graphs

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text]. An edge subset [Formula: see text] is called a restricted edge-cut if [Formula: see text] is disconnected and has no isolated vertices. The restricted edge-connectivity [Formula: see text] of [Formula: see text] is the cardinality of a minimum restricted edge-cut of [Formula: see text] if it has any; otherwise [Formula: see text]. If [Formula: see text] is not a star and its order is at least four, then [Formula: see text], where [Formula: see text]. The graph [Formula: see text] is said to be maximally restricted edge-connected if [Formula: see text]; the graph [Formula: see text] is said to be super restricted edge-connected if every minimum restricted edge-cut isolates an edge from [Formula: see text]. The direct product of graphs [Formula: see text] and [Formula: see text], denoted by [Formula: see text], is the graph with vertex set [Formula: see text], where two vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. In this paper, we give a sufficient condition for [Formula: see text] to be super restricted edge-connected, where [Formula: see text] is the complete graph on [Formula: see text] vertices.

Connectivity and Nature Diagnosability of Leaf-Sort Graphs

The connectivity and diagnosability of a multiprocessor system or an interconnection network is an important research topic. The system and interconnection network has a underlying topology, which usually presented by a graph. As a famous topology structure of interconnection networks, the n-dimensional leaf-sort graph CFn has many good properties. In this paper, we prove that (a) the restricted edge connectivity of CFn (n ≥ 3) is 3n − 5 for odd n and 3n − 6 for even n; (b) CFn (n ≥ 5) is super restricted edge-connected; (c) the nature diagnosability of CFn (n ≥ 4) under the PMC model is 3n − 4 for odd n and 3n − 5 for even n; (d) the nature diagnosability of CFn (n ≥ 5) under the MM* model is 3n − 4 for odd n and 3n − 5 for even n.

The Edge Connectivity of Expanded k-Ary n-Cubes

Mass data processing and complex problem solving have higher and higher demands for performance of multiprocessor systems. Many multiprocessor systems have interconnection networks as underlying topologies. The interconnection network determines the performance of a multiprocessor system. The network is usually represented by a graph where nodes (vertices) represent processors and links (edges) represent communication links between processors. For the network G, two vertices u and v of G are said to be connected if there is a (u,v)-path in G. If G has exactly one component, then G is connected; otherwise G is disconnected. In the system where the processors and their communication links to each other are likely to fail, it is important to consider the fault tolerance of the network. For a connected network G=(V,E), its inverse problem is that G-F is disconnected, where F⊆V or F⊆E. The connectivity or edge connectivity is the minimum number of F. Connectivity plays an important role in measuring the fault tolerance of the network. As a topology structure of interconnection networks, the expanded k-ary n-cube XQnk has many good properties. In this paper, we prove that (1) XQnk is super edge-connected (n≥3); (2) the restricted edge connectivity of XQnk is 8n-2 (n≥3); (3) XQnk is super restricted edge-connected (n≥3).

On restricted edge-connectivity of lexicographic product graphs

Let G1 be a nontrivial connected graph and G2 be a nonempty graph. Denote by G1° G2 the lexicographic product of G1 and G2. Let λ′(G1° G2) and ξ(G1° G2) represent the restricted edge-connectivity and the minimum edge-degree of G1° G2, respectively. In this paper, we show that , where ni, λi, δi and ξi are the order, the edge-connectivity, the minimum degree and the minimum edge-degree of Gi (i=1, 2) respectively. Moreover, when the order of G2 is at least five, we prove that: (i) G1° G2 is super restricted edge-connected if and only if ; (ii) if G1 is maximally edge-connected, then G1° G2 is super restricted edge-connected; (iii) if G1 is maximally restricted edge-connected, then G1° G2 is super restricted edge-connected.

Super restricted edge-connectivity of graphs with diameter 2

For a connected graph G, an edge-cut S is called a restricted edge-cut if G−S contains no isolated vertices. And G is said to be super restricted edge-connected, for short super-λ′, if each minimum restricted edge-cut of G isolates an edge. Let Vδ denote the set of the minimum degree vertices of G. In this paper, for a super-λ′ graph G with diameter D≥2 and minimum degree δ≥4, we show that the induced subgraph G[Vδ] contains no complete graph Kδ−1. Applying this property we characterize the super restricted edge connected graphs with diameter 2 which satisfy a type of neighborhood condition. This result improves the previous related one which was given by Wang et al. [S. Wang, J. Li, L. Wu, S. Lin, Neighborhood conditions for graphs to be super restricted edge connected, Networks 56 (2010) 11–19].

Super restricted edge connectivity of regular graphs with two orbits

It is well known that the edge-connectivity of a simple, connected, vertex transitive graph attains its regular degree. It is then natural to consider the relationship between the graph’s edge connectivity and the number of orbits of its automorphism group. In [6], Liu and Meng (2008) studied the edge connectivity of regular double-orbits graphs. Later, Lin et al. (in press) [10] characterized the λ′-optimal 3-regular double-orbit graph and given a sufficient condition for the k-regular double-orbit graphs to be optimal. In this note, we characterize the super restricted edge connected k-regular double-orbit graphs with grith at least 6.

Super restricted edge connectivity of regular edge-transitive graphs

An edge cut of a connected graph is called restricted if it separates this graph into components each having order at least 2; a graph G is super restricted edge connected if G−F contains an isolated edge for every minimum restricted edge cut F of G. It is proved in this paper that a connected regular edge-transitive graph of valency at least 3 is not super restricted edge connected if and only if it is either the three dimensional hypercube, or a tetravalent edge-transitive graph of girth 3 and of order at least 6. As a result, there are infinitely many k-regular Hamiltonian graphs with k≥3 which are not super restricted edge connected. This answers negatively a question in [J. Ou, F. Zhang, Super restricted edge connectivity of regular graphs, Graphs & Combin. 21 (2005) 459–467] regarding the relationship between restricted edge connected graphs and Hamiltonian graphs.

Neighborhood conditions for graphs to be super restricted edge connected

AbstractRestricted edge connectivity is a more refined network reliability index than edge connectivity. For a connected graph G = (V, E), an edge set S ⊆ E is a restricted edge cut if G − S is disconnected and every component of G − S has at least two vertices. The restricted edge connectivity of G is defined as the cardinality of a minimum restricted edge cut. G is super restricted edge connected if every minimum restricted edge cut of G isolates one edge. In this article, we present several neighborhood conditions for a graph to be super restricted edge connected. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010

On super restricted edge-connectivity of edge-transitive graphs

Let X = ( V , E ) be a connected graph. Then X is called super restricted edge-connected, in short, sup- λ ′ , if every minimum edge set F of X such that X − F is disconnected and every component of X − F has at least two vertices, the set of edges being adjacent to a certain edge with minimum edge degree in X . In this paper, by studying the λ ′ -superatoms of X , we characterize super restricted edge-connected edge-transitive graphs. As a corollary, we present a sufficient and necessary condition for connected vertex-transitive line graphs to be super vertex-connected.

Super restricted edge connected Cartesian product graphs

An edge-cut F of a connected graph G is called a restricted edge-cut if G − F contains no isolated vertices. The minimum cardinality of all restricted edge-cuts is called the restricted edge-connectivity λ ′ ( G ) of G. A graph G is said to be λ ′ -optimal if λ ′ ( G ) = ξ ( G ) , where ξ ( G ) is the minimum edge-degree of G. A graph is said to be super- λ ′ if every minimum restricted edge-cut isolates an edge. This article gives a sufficient condition for Cartesian product graphs to be super- λ ′ . Using this result, certain classes of networks which are recursively defined by the Cartesian product can be simply shown to be super- λ ′ .

Sufficient conditions for bipartite graphs to be super- [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 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 U 1 , U 2 ⊂ 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 ξ k ( G ) = min { | [ U , V ( G ) ∖ U ] | : U ⊂ V ( G ) , | U | = k ≥ 1 and the subgraph induced by U is connected } . A graph G is λ k -optimal if λ k ( G ) = ξ k ( G ) . Furthermore, if every minimum k -restricted edge cut is a set of edges incident to a certain connected subgraph of order k , then G is said to be super- k -restricted edge connected or super- λ k for simplicity. Let k be a positive integer and let G be a bipartite graph of order n ≥ 4 with the bipartition ( X , Y ) . In this paper, we prove that: (a) If G has a matching that saturates every vertex in X or every vertex in Y and | N ( u ) ∩ N ( v ) | ≥ 2 for any u , v ∈ X and any u , v ∈ Y , then G is λ 2 -optimal; (b) If G has a matching that saturates every vertex in X or every vertex in Y and | N ( u ) ∩ N ( v ) | ≥ 3 for any u , v ∈ X and any u , v ∈ Y , then G is super- λ 2 ; (c) If the minimum degree δ ( G ) ≥ n + 2 k 4 , then G is λ k -optimal; (d) If the minimum degree δ ( G ) ≥ n + 2 k + 3 4 , then G is super- λ k .

Minimally restricted edge connected graphs

For a connected graph G = ( V , E ) , an edge set S ⊂ E is a restricted edge cut if G − S is disconnected and there is no isolated vertex in G − S . The cardinality of a minimum restricted edge cut of G is the restricted edge connectivity of G , denoted by λ ′ ( G ) . A graph G is called minimally restricted edge connected if λ ′ ( G − e ) < λ ′ ( G ) for each edge e ∈ E . A graph G is λ ′ -optimal if λ ′ ( G ) = ξ ( G ) , where ξ ( G ) is the minimum edge degree of G . We show in this work that a minimally restricted edge connected graph is always λ ′ -optimal.

The maximal restricted edge connectivity of Kautz undirected graphs

Abstract The restricted edge connectivity is an accurate measure of fault-tolerance for interconnection networks. In this paper, we obtain the following: for d ≥ 3 , n ≥ 2 , the restricted edge connectivity of Kautz undirected graph U K ( d , n ) is 4 d − 4 , and U K ( d , n ) is a maximal restricted edge connected graph.

Super restricted edge-connectivity of vertex-transitive graphs

We prove that a connected vertex-transitive graph G with degree k > 2 and girth g > 4 is super restricted edge-connected, that is, if F is a set of 2 k - 2 edges of G such that G - F is disconnected and every component of G - F has at least two vertices, then F is the set of edges adjacent to a certain edge in G. We also show that neither the condition k > 2 nor g > 4 can be weakened.

A Graph-Theoretic Approach to a Communications Problem

A Graph-Theoretic Approach to a Communications Problem