### Paths in Graphs and Curves on Surfaces [chapter]

Alexander Schrijver
<span title="">1994</span> <i title="Birkhäuser Basel"> First European Congress of Mathematics Paris, July 6–10, 1992 </i> &nbsp;
A classical result in graph theory due to the topologist Menger [24] concerns the existence of pairwise internally vertex-disjoint paths connecting two given vertices r and s in an undirected graph G = (V, E). (Two paths are internally vertex-disjoint if they do not have any vertex or edge in common, except for the end vertices.) Menger's theorem states that the maximum number of such paths is equal to the minimum size of a set W of vertices with r, s rJ. W such that each path from r to s
more &raquo; ... ects W at least once. (It is assumed that r and s are not adjacent.) The interest of Menger in this problem originated from characterizing the bifurcation number of certain topological spaces which he named K urven. A similar theorem holds if we replace 'undirected' by 'directed'. Moreover, variants are obtained by replacing 'internally vertex-disjoint' by 'edgedisjoint'. (Two paths are edge-disjoint if they do not have any edge in common.) Menger's theorem is a basic result in graph theory, and several other theorems in graph theory utilize Menger's theorem in some way. Application in optimization followed when Ford and Fulkerson [6] proved their famous max-fiow min-cut theorem: the maximum amount of 'flow' that can be transmitted from some 'source' r to some 'sink' s in a capacitated network, is equal to the minimum capacity of any r -s cut. (An r -s cut is a set of edges intersecting each path from r to s.) This theorem can be derived from Menger's theorem (and vice versa). Ford and Fulkerson [7] also designed a fast algorithm to determine a maximum flow from r to s. This method formed the basis for a wealth of applications in operations research, e.g., to problems involving assignment, transportation, transshipment, routing, circulation, and communication. Menger's theorem and the max-fiow min-cut theorem are of interest also because they provide us with a way of 'certifying' that a certain path packing or a certain flow is the largest possible. Indeed, to convince somebody that a certain collection of k pairwise internally vertex-disjoint r -s paths is the largest possible, it suffices to exhibit a set W of vertices intersecting each r -s path and satisfying \W\ = kby Menger's theorem that such a set always exists. In the language of complexity theory, this CWI,
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