For a connected graph the restricted edge-connectivity λ (G) is defined as the minimum cardinality of an edge-cut over all edge-cuts S such that there are no isolated vertices in (2004), 113-120] gave a sufficient condition for λ -optimality in graphs of diameter 2. In this paper, we generalize this condition in graphs of diameter g − 1, g being the girth of the graph, and show that a graph G with diameter at most g − 2 is λ -optimal.

The product graph G m * G p of two given graphs G m and G p was defined by Bermond et al. [Large graphs with given degree and diameter II, J. Combin. Theory Ser. B 36 (1984) 32-48]. For this kind of graphs we provide bounds for two connectivity parameters ( and , edge-connectivity and restricted edge-connectivity, respectively), and state sufficient conditions to guarantee optimal values of these parameters. Moreover, we compare our results with other previous related ones for permutation graphs and cartesian product graphs, obtaining several extensions and improvements. In this regard, for any two connected graphs G m ,

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The product graph G m * G p of two given graphs G m and This work deals with product graphs for which we provide bounds for the connectivity parameter κ. Moreover, we state sufficient conditions that guarantee these product graphs to be maximally connected or superconnected. As a consequence, we deduce that even small networks with low reliability may lead to larger networks with high levels of fault-tolerance.

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.

"Discrete Mathemetics Top Cited Article 2005-2010"The restricted connectivity κ′(G)κ′(G) of a connected graph G is defined as the minimum cardinality of a vertex-cut over all vertex-cuts X such that no vertex uu has all its neighbors in X; the superconnectivity κ1(G)κ1(G) is defined similarly, this time considering only vertices uu in G-XG-X, hence κ1(G)⩽κ′(G)κ1(G)⩽κ′(G). The minimum edge-degree of G is ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}, d(u)d(u) standing for the degree of a vertex uu. In this paper, several sufficient conditions yielding κ1(G)⩾ξ(G)κ1(G)⩾ξ(G) are given, improving a previous related result by Fiol et al. [Short paths and connectivity in graphs and digraphs, Ars Combin. 29B (1990) 17–31] and guaranteeing κ1(G)=κ′(G)=ξ(G)κ1(G)=κ′(G)=ξ(G) under some additional constraints.Peer ReviewedAward-winningPostprint (published version

For a strongly connected digraph D the restricted arc‐connectivity λ′(D) is defined as the minimum cardinality of an arc‐cut over all arc‐cuts S satisfying that D ‐ S has a non‐trivial strong component D1 such that D ‐ V (D1) contains an arc. In this paper we prove that every digraph on at least 4 vertices and of minimum degree at least 2 is λ′ ‐connected and λ′(D) ≤ξ′(D), where ξ′(D) is the minimum arc‐degree of D. Also in this paper we introduce the concept of super‐ λ′ digraphs and provide a sufficient condition for an s ‐geodetic digraph to be super‐ λ′. Further, we show that the h ‐iterated line digraph Lh(D) of an s ‐geodetic digraph is super‐ λ′ for a particular h. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013

A 3 − arc of a graph G is a 4-tuple (y, a, b, x) of vertices such that both (y, a, b) and (a, b, x) are paths of length two in G.