On restricted connectivities of permutation graphs

• Authors: C. Balbuena;X. Marcote;P. García-Vázquez
• Venue: Networks
• Year: 2005

A permutation graph (or generalized prism)Gπ of a graphG is obtained by taking two disjoint copies ofG and adding an arbitrary matching between thetwo copies. Permutation graphs can be seen as suitable models forbuilding larger interconnection networks from smaller ones withoutincreasing significantly their maximum transmission delays, in sucha way that these larger networks are highly fault-tolerant. Forpermutations graphs, in this article we provide conditions thatguarantee optimal values for two parameters of connectivity,λ′ and κ′. For a connected graphG the restricted edge-connectivityλ′(G) is definedas the minimum cardinality of a restricted edge-cut; that is, theminimum cardinality of a set S of edges suchthat G - S is not connectedand S does not contain the set of incident edgesof any vertex of the graph. A graph G is said tobe λ′-optimal ifλ′(G)= ξ(G), whereξ(G) is the minimumedge-degree in G defined asξ(G) =min{d(u)+d(v) - 2 :uv ∈E(G)},and d(u)denotes the degree of vertex u. Among otherthings, we prove that permutation graphs satisfy:min{λ′(G)+δ(G),2λ′(G),ξ(Gπ)}≤λ′(Gπ)≤ξ(Gπ)if|V(G)| ≥ ξ(G)+ 2. Furthermore, min{2λ′(G),ξ(Gπ)}≤λ′(Gπ)≤ξ(Gπ)if G is triangle-free. We also study the vertexcase considering the restricted connectivityκ′(G) and relatingit to the superconnectivityκ1(G);the latter is defined as the minimum cardinality of a set ofvertices, if any, whose deletion disconnects Gin such a way that every remaining component has at least twovertices. For instance, we prove that2κ(G) ≤κ1(G)≤κ′(Gπ)≤ξ(Gπ)if G is triangle-free and the permutation graphhas no cycles of length five. © 2005 Wiley Periodicals, Inc.NETWORKS, Vol. 45(3), 113–118 2005