On restricted connectivities of permutation graphs

Abstract

A permutation graph (or generalized prism) G π of a graph G is obtained by taking two disjoint copies of G and adding an arbitrary matching between the two copies. Permutation graphs can be seen as suitable models for building larger interconnection networks from smaller ones without increasing significantly their maximum transmission delays, in such a way that these larger networks are highly fault-tolerant. For permutations graphs, in this article we provide conditions that guarantee optimal values for two parameters of connectivity, λ and κ . For a connected graph G the restricted edge-connectivity λ (G) is defined as the minimum cardinality of a restricted edge-cut; that is, the minimum cardinality of a set S of edges such that G − S is not connected and S does not contain the set of incident edges of any vertex of the graph. A graph G is said to be λ -optimal if λ (G) = ξ (G), where ξ (G) is the minimum edge-degree in G defined as ξ (G) = min{d(u) + d(v) − 2 : uv ∈ E(G)}, and d(u) denotes the degree of vertex u. Among other things, we prove that permutation graphs satisfy:if G is triangle-free. We also study the vertex case considering the restricted connectivity κ (G) and relating it to the superconnectivity κ 1 (G); the latter is defined as the minimum cardinality of a set of vertices, if any, whose deletion disconnects G in such a way that every remaining component has at least two vertices. For instance, we prove that 2κ(G) ≤ κ 1 (G) ≤ κ (G π ) ≤ ξ (G π ) if G is triangle-free and the permutation graph has no cycles of length five.

Sufficient conditions for graphs to be λ′‐optimal and super‐λ′

“…Since G is λ -connected, if G is non-λ 3 -connected, then every restricted edge-cut isolates an edge of G and G is super-λ . Now assume that S = [X,X] is a λ 3 Then, λ 3 (G) > ξ(G) ≥ λ (G). Hence every λ -cut of G isolates an edge, that is, G is super-λ .…”

“…Balbuena et al [2] generalized the conditions of [9] to graphs of diameter g−1, where g is the girth of the graph. In addition, several authors have studied different kinds of graphs with λ -optimality, such as Latifi et al [11] and Wu and Guo [22] for hypercubes, Li and Li [12] for circulant graphs, Ueffing and Volkmann [17] for Cartesian product of graphs, Wang and Li [21] for graphs of diameter 2, Balbuena et al for permutation graphs [3] and graphs with small conditional diameter [1], and Xu [23], Xu and Xu [24], and Meng [13] for edge-transitive and vertex-transitive graphs, etc.…”

On super connectivity of Cartesian product graphs

“…The restricted connectivity κ (G) and super connectivity κ 1 (G) are related [2,4] and have received the attention of many researchers (see, for example, [1,9,14,15,19]). Obviously, κ 1 (G) ≤ κ (G) for any restricted vertex-cut S is a super vertex-cut.…”

Reliability of interconnection networks modeled by Cartesian product digraphs

“…For a graph G and a permutation π of V (G), the permutation graph G π is defined by taking two disjoint copies of G and adding a matching joining each vertex v in the first copy to π(v) in the second copy. The connectivity and edge-connectivity of G π were studied in [2,5,7]. It is clear that permutation graphs cannot contain the Cartesian product graphs.…”