Krishnamoorthy, Thulasiraman and Swamy [Minimum order graphs with specified diameter, connectivity and regularity, Networks 19 (1989) 25–46] showed that a δ-regular graph with diameter D at most 3 has (vertex-)connectivity κ at least 2, and if D ≤ 2 then the connectivity is at least κ ≥ min{δ, 3}. Likewise, Soneoka, Nakada, Imase and Peyrat [Sufficient conditions for maximally connected graphs,...

A subset T ⊆ V of terminals is k-connected to a root s in a directed/undirected graph J if J has k internally-disjoint vs-paths for every v ∈ T ; T is k-connected in J if T is k-connected to every s ∈ T . We consider the Subset k-Connectivity Augmentation problem: given a graph G = (V,E) with edge/node-costs, node subset T ⊆ V , and a subgraph J = (V,EJ ) of G such that T is k-connected in J , ...

Vertex and edge connectivity are special cases of mixed connectivity, in which all edges and a specified set of vertices play a similar role. Certificates of k-connectivity for a graph are obtained by removing a subset of its edges, while preserving its connectivity up to k. We unify the previous work on connectivity certificates and extend it to handle mixed connectivity and multigraphs. Our t...

Let (X,R) be a commutative association scheme and let Γ = (X,R ∪R>) be a connected undirected graph where R ∈ R. Godsil (resp., Brouwer) conjectured that the edge connectivity (resp., vertex connectivity) of Γ is equal to its valency. In this paper, we prove that the deletion of the neighborhood of any vertex leaves behind at most one non-singleton component. Two distinct vertices a, b ∈ X are ...

A connected graph is said to be super edge-connected if every minimum edge-cut isolates a vertex. The restricted edge-connectivity λ′ of a connected graph is the minimum number of edges whose deletion results in a disconnected graph such that each connected component has at least two vertices. A graph G is called λ′-optimal if λ′(G) = min{dG(u)+dG(v)−2 : uv is an edge in G}. This paper proves t...

A (δ, g)-cage is a regular graph of degree δ and girth g with the least possible number of vertices. Recently, some authors have addressed the problem of studying their connectivity parameters. In this direction, it was conjectured by Fu, Huang and Rodger that every (δ, g)-cage is maximally connected, i.e., it is δ-connected, and they proved this statement for δ = 3. We provide a new contributi...

We introduce a new class of fixed-degree interconnection networks, called the Cayley graph connected cycles , which includes the well known cube-connected cycles as a special case. This class of networks is shown to be vertexsymmetric and maximally fault tolerant (if the given Cayley graph is maximally fault tolerant). We propose simple routing and broadcasting algorithms for these networks in ...

Article history: Received 8 August 2008 Received in revised form 21 January 2012 Accepted 8 March 2014 Available online 19 March 2014