Minimax theorems for infinite graphs with the ends as ideal points
For a family \({\mathcal A}= (A_ i)_{i\in I}\) of disjoint sets of ends in a graph \(G= (V, E)\) the author proves Menger-type results involving the concepts of an \({\mathcal A}\)-path and of an \({\mathcal A}\)-separator. By definition, an \({\mathcal A}\)-path joins elements of distinct \(A_ i\) and \(A_ j\), while a path \(P\), finite or infinite, joins the distinct ends \(\tau_ 1\) and \(\tau_ 2\), if \(\tau_ h\) corresponds to a ray \(R_ h\subseteq P\) with \(R_ h\in \tau_ h\) or else to a terminal vertex \(v_ h\) of \(P\) which then cannot be separated from \(v_ h\) by a finite set \(F\subseteq V- \{v_ h\}\) (\(h= 1,2\); \(v_ 1\neq v_ 2\), if \(P\) is finite). A set \(M\) of vertices and edges separates \(\mathcal A\) if each \({\mathcal A}\)-path contains an element of \(M\). By definition, such a set is dispersed by \(\mathcal A\) if for each \(\tau\in \bigcap {\mathcal A}\) some finite set \(F\subseteq V\) separates \(M\) from \(\tau\) (in the usual sense). By the two main results the maximum cardinality of sets of disjoint \({\mathcal A}\)- arcs equals the minimum cardinality of certain \({\mathcal A}\)-separating sets which are dispersed by \(\mathcal A\). Special attention is given to the case when \(| I|= 2\).
scientific article; zbMATH DE number 626314
scientific article