On restricted connectivities of permutation graphs

A permutation graph (or generalized prism) G 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: min{λ′(G) + δ(G), 2λ′(G), ξ(G )} ≤ λ′(Gπ ) ≤ ξ(G ) if |V(G)| ≥ ξ(G) + 2. Furthermore, min{2λ′(G), ξ(G )} ≤ λ′(Gπ ) ≤ ξ(G ) if G is triangle-free. We also study the vertex case considering the restricted connectivity κ ′(G) and relating it to the superconnectivity κ1(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) ≤ κ1(G) ≤ κ ′(Gπ ) ≤ ξ(G ) if G is triangle-free and the permutation graph has no cycles of length five. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 45(3), 113–118 2005