Given a graph G, a non-negative integer g and a set of vertices S, the g-extra connectivity of G is the cardinality of a minimum set S such that G − S is disconnected and each component of G− S has at least g + 1 vertices. The 2-extra connectivity of k-ary n-cubes is gotten by Hsieh et al. in [Theoretical Computer Science, 443 (2012) 63-69] for k ≥ 4. This paper shows that the 3-extra connectiv...

Fuzzy incidence graphs can be used as models for nondeterministic interconnection networks having extra node-edgerelationships. For example, ramps in a highway system may be modeled as a fuzzy incidence graph so that unexpectedflow between cities and highways can be effectively studied and controlled. Like node and edge connectivity in graphs,node connectivity and arc connectivity in fuzzy inci...

In a 1973 paper, Cooke obtained an upper bound on the possible connectivity of a graph embedded in a surface (orientable or nonorientable) of fixed genus. Furthermore, he claimed that for each orientable genus #>0 (respectively, nonorientable genus # >0, # {2) there is a complete graph of orientable genus # (respectively, nonorientable genus # ) and having connectivity attaining his bound. It i...

Let G be a connected graph and S be a set of vertices. The h-extra connectivity of G is the cardinality of a minimum set S such that G − S is disconnected and each component of G− S has at least h + 1 vertices. The h-extra connectivity is an important parameter to measure the reliability and fault tolerance ability of large interconnection networks. The h-extra connectivity for h = 1, 2 of k-ar...

Let P be a Poisson process of intensity one in a square Sn of area n. We construct a random geometric graph Gn,k by joining each point of P to its k nearest neighbours. For many applications it is desirable that Gn,k is highly connected, that is, it remains connected even after the removal of a small number of its vertices. In this paper we relate the study of the s-connectivity of Gn,k to our ...

Let $R$ be a ring (not necessarily commutative) with nonzero identity. We define $Gamma(R)$ to be the graph with vertex set $R$ in which two distinct vertices $x$ and $y$ are adjacent if and only if there exist unit elements $u,v$ of $R$ such that $x+uyv$ is a unit of $R$. In this paper, basic properties of $Gamma(R)$ are studied. We investigate connectivity and the girth of $Gamma(R)$, where $...

Directed s-t connectivity is the problem of detecting whether there is a path from vertex s to vertex t in a directed graph. We present the rst known deterministic sublinear space, polynomial time algorithm for directed s-t connectivity. For n-vertex graphs, our algorithm can use as little as n=2 ( p logn) space while still running in polynomial time.

Let G = (V,E) be an undirected graph and let S ⊆ V . The S-connectivity SG(u, v) of a pair of nodes u, v in G is the maximum number of uv-paths that no two of them have an edge or a node in S ∖ {u, v} in common. Edge-connectivity is the case S = ∅ and node-connectivity is the case S = V . Given a graph G = (V,E), an integer k, a subset T ⊆ V of terminals, and S ⊆ V , we consider the problem of ...