3-restricted connectivity of graphs with given girth

Abstract

Let G = (V, E) be a connected graph. X ⊂ V (G) is a vertex set. X is a 3-restricted cut of G, if G−X is not connected and every component of G−X has at least three vertices. The 3-restricted connectivity κ3(G) (in short κ3) of G is the cardinality of a minimum 3-restrictedA network is often modelled by a graph G = (V, E) with the vertices representing nodes such as processors or stations, and the edges representing links between the nodes. Throughout this paper, we assume the graphs considered are simple.Let d(u, v) denotes the length of a shortest (u, v)-path. If X, Y ⊂ V , d(X, Y ) = min{d(x, y) : for any x ∈ X and any y ∈ Y } denotes the distance between X and Y . v ∈ V, r ≥ 0 is an integer,is the subgraph induced by X. We denote the diameter and girth by D and g, respectively, and write G − v for G − {v}. A path is called k-path, if its length of edges is k.An edge set S is called a k-restricted edge cut of G, if G − S is not connected and every connected component of G−S has at least k vertices. The cardinality of a minimum k-restricted edge cut is the k-restricted edge connectivity of G, denoted by λ k (G). As for the recent studies in this aspect, we can see [2][3][4][5][6][7][8][9].