Let G be a nontrivial connected graph of order n and let k be an integer with 2 ≤ k ≤ n. For a set S of k vertices of G, let κ(S) denote the maximum number ℓ of edge-disjoint treeswith this property is called an internally disjoint set of trees connecting S. Chartrand et al. generalized the concept of connectivity as follows: The k-connectivity, denoted by κ k (G), of G is defined by κ k (G) =min{κ(S)}, where the minimum is taken over all k-subsets S of V (G). Thus κ 2 (G) = κ(G), where κ(G) is the connectivity of G.In general, the investigation of κ k (G) is very difficult. We therefore focus on the investigation on κ 3 (G) in this paper. We study the relation between the connectivity and the 3-connectivity of a graph. First we give sharp upper and lower bounds of κ 3 (G) for general graphs G, and construct two kinds of graphs which attain the upper and lower bound, respectively. We then show that if G is a connected planar graph, then κ(G) − 1 ≤ κ 3 (G) ≤ κ(G), and give some classes of graphs which attain the bounds. In the end we show that the problem whether κ(G) = κ 3 (G) for a planar graph G can be solved in polynomial time.

No abstract

SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians.More information about this series at

Let G be a graph, S be a set of vertices of G, and λ(S) be the maximum numberIn this paper, we consider the Nordhaus-Gaddum-type results for the parameter λ k (G). We determine sharp upper and lower bounds of λ k (G) + λ k (G) and λ k (G) · λ k (G) for a graph G of order n, as well as for a graph of order n and size m. Some graph classes attaining these bounds are also given.