Some extremal connectivity results for matroids
\textit{P. D. Seymour} [J. Comb. Theory, Ser. B 26, 159-173 (1979; Zbl 0443.05029)] proved that the 4-point line is the only connected matroid with more than one element in which every 2-element deletion and every 2- element contraction is disconnected, but every 1-element deletion and every 1-element contraction is connected. This paper proves the following result from which one can derive a generalization of Seymour's theorem. Let M be a matroid having at least \(n+2\) elements where n is an integer exceeding one. Then every n-element subset X of E(M) is in an \((n+1)\)- element circuit if and only if (i) for every such subset M/X is disconnected, and (ii) for every smaller subset Y, M/Y is connected. Among the other consequences of this theorem that are obtained is a characterization, in terms of a connectivity condition, of Sylvester matroids, a class of matroids introduced by \textit{U. S. R. Murty} [Aequationes Math. 4, 44-50 (1970; Zbl 0196.027)].
scientific article; zbMATH DE number 4123747
scientific article