Mengerian theorems for paths of bounded length

Abstract

Let u and v be non-adjacent points in a connected graph G . A classical result known to all graph theorists is that called MENGER's theorem . The point version of this result says that the maximum number of point-disjoint paths joining u and v is equal to the minimum number of points whose deletion destroys all paths joining u and v . The theorem may be proved purely in the language of graphs (probably the best known proof is indirect, and is due to DiRAC [3] while a more neglected, but direct, proof may be found in ORE [7]) . One may also prove the theorem by appealing to flow theory (e.g . BERGE [1], p. 167) . In many real-world situations which can be modeled by graphs certain paths joining two non-adjacent points may well exist, but may prove essentially useless because they are too long . Such considerations led the authors to study the following two parameters . Let n be any positive integer and let u and v be any two non-adjacent points in a graph G . Denote by A„(u, v) the maximum number of point-disjoint paths joining u and v whose length (i .e ., number of lines) does not exceed n. Analogously, let V„(u, v) be the minimum number of points in G the deletion of which destroys all paths joining u and v which do not exceed n in length., A special case would obtain when n = p = I V(G)I, and we have by Monger's theorem, the equality A„(u, v) = V,,(u, v) .