Critically «connected Graphs

The following result is proved. Every «-connected graph contains either a vertex whose removal results in a graph which is also «-connected or a vertex of degree less than (3n—1)/2. Introduction. A graph G is said to be n-connected if the removal of fewer than « vertices from G neither disconnects it nor reduces it to the trivial graph consisting of a single vertex. The maximum value of « for which a graph G is «-connected is called its connectivity and is denoted by k(G). The minimum degree of G is designated by (3(G); the inequality /c(G)_ó(G) is well known. A graph G is said to be critically n-connected if k(G)=u and k(G—v)= n—\ for each vertex v of G. Analogously, a graph G is minimally nconnected if x(G)=n and for each edge e of G, «-(G—e)=n—1. The object of this article is to present a necessary condition for a graph to be critically «-connected and to discuss related topics. Since 1-connected graphs are the nontrivial connected graphs and since every nontrivial connected graph G has at least two vertices u and v such