Menger's Theorem for digraphs states that for any two vertex sets A and B of a digraph D such that A cannot be separated from B by a set of at most t vertices, there are t + 1 disjoint A-B-paths in D. Here a short and elementary proof of a more general theorem is given. For terminology and notation not deened here we refer to 1]. Recall that for a directed… (More)
Let F be a set of graphs and for a graph G let α F (G) and α * F (G) denote the maximum order of an induced subgraph of G which does not contain a graph in F as a subgraph and which does not contain a graph in F as an induced subgraph, respectively. Lower bounds on α F (G) and α * F (G) and algorithms realizing them are presented.
For a set X of vertices of a graph fulfilling local connectedness conditions the existence of a cycle containing X is proved.
For c 2 and k minfc; 3g, guaranteed upper bounds on the length of a shortest cycle through k prescribed vertices of a c-connected graph are proved. Analogous results on planar graphs are presented, too.
In analogy to the absolute algebraic connectivity of Fiedler, we study the problem of minimizing the maximum eigenvalue of the Laplacian of a graph by redistributing the edge weights. Via semidefinite duality this leads to a graph realization problem in which nodes should be placed as close as possible to the origin while adjacent nodes must keep a distance… (More)
A short proof of the classical theorem of Menger concerning the number of disjoint AB-paths of a finite graph for two subsets A and B of its vertex set is given. The main idea of the proof is to contract an edge of the graph. Proofs of Menger's Theorem are given in [7, 6, 4, 8, 2]. A short proof is given by T. Böhme, F. Göring and J. Harant in ; another… (More)