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

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

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