For a 2-connected graph G and vertices u, v of G we define an abstract graph P(Guv) whose vertices are the paths joining u and v in G, where paths S and T are adjacent if T is obtained from S by replacing a subpath Sxy of S with an internally disjoint subpath Txy of T . We prove that P(Guv) is always connected and give a necessary and a sufficient condition for connectedness in cases where the cycles formed by the replacing subpaths are restricted to a specific family of cycles of G.

The perfect matching graph of a graph G, denoted by M(G), has one vertex for each perfect matching of G and two matchings are adjacent if their symmetric difference is a cycle of G. Let C be a family… Expand

Abstract. We investigate the vertex-connectivity of the graph of f-monotone paths on a d-polytopeP with respect to a generic functionalf. The third author has conjectured that this graph is always (d… Expand

It is shown that any tree graph containing more than two vertices has Property H, and two operations for augmenting networks (linear graphs) are defined: edge insertion and vertex insertion.Expand

Every generic linear functional $f$ on a convex polytope $P$ orients the edges of the graph of $P$. In this directed graph one can define a notion of $f$-arborescence and $f$-monotone path on $P$.… Expand