Characterizations of n-connected and n-line connected graphs

Abstract

The following two theorems are proved: (1) A graph G with at least n + 1 points, n ≥ 2, is n-connected if and only if for each set S of n distinct points of G and for each two point subset T of S there is a cycle of G that contains the points of T and avoids the points of S − T. (2) A graph G with at least n + 1 lines, n ≥ 2, with no isolated points is n-line connected if and only if for each set S of n distinct lines of G and for each two line subset T of S there is a circuit of G that contains the lines of T and avoids the lines of S − T.