Diameter-girth sufficient conditions for optimal extraconnectivity in graphs

For a connected graph G, the r-th extraconnectivity κr(G) is defined as the minimum cardinality of a cutset X such that all remaining components after the deletion of the vertices of X have at least r+1 vertices. The standard connectivity and superconnectivity correspond to κ0(G) and κ1(G), respectively. The minimum r-tree degree of G, denoted by ξr(G), is the minimum cardinality of N(T ) taken over all trees T ⊆ G of order |V (T )| = r + 1, N(T ) being the set of vertices not in T that are neighbors of some vertex of T . When r = 1, any such considered tree is just an edge of G. Then, ξ1(G) is equal to the so called minimum edge-degree of G, defined as ξ(G) = min{d(u) + d(v) − 2 : uv ∈ E(G)}, where d(u) stands for the degree of vertex u. A graph G is said to be optimally r-extraconnected, for short κr-optimal, if κr(G) ≥ ξr(G). In this paper, we present some sufficient conditions that guarantee κr(G) ≥ ξr(G) for r ≥ 2. These results improve some previous related ones, and can be seen as a complement of some others which were obtained by the authors for r = 1.