On finding convex cuts in general, bipartite and plane graphs

Abstract The general notion of convexity also applies to a graph G = ( V , E ) . A subset V c of V is called a convex set if all shortest paths with end vertices in V c remain within V c . A convex cut of G is a partition ( V 1 , V 2 ) of V such that V 1 and V 2 are convex sets (halfspaces). Finding convex cuts is NP -hard for general graphs. In this paper we first characterize the convex cuts of a connected bipartite graph G ′ in terms of the Djokovic relation, a reflexive and symmetric relation on the edges of a graph that is based on shortest paths between the edges' end vertices. Specifically, we show that a cut of G ′ is convex if and only if the Djokovic relation holds for any pair of edges in its cut-set. As a consequence, all convex cuts of G ′ = { V ′ , E ′ } can be found in O ( | V ′ | | E ′ | ) . We then characterize the convex cuts of a general connected graph G using the Djokovic–Winkler relation θ and another relation τ on the edges of G. Based on this general characterization, we show how one can find all convex cuts of G in polynomial time. The key parts here are (i) describing the Djokovic–Winkler relation on the edges of G in terms of the Djokovic relation θ ′ on the bipartite graph obtained from G by subdividing each edge of G into two edges, (ii) studying the interplay of θ ′ and τ on plane curves representing convex cuts, and (iii) a running time analysis of our algorithm for finding the convex cuts. Our method for characterizing and finding convex cuts of a connected plane graph G is motivated by the concept of alternating cuts and conditions on the latter to be convex. In the last part of this paper we represent alternating cuts as plane curves and focus on their intersection pattern. If the plane curves form an arrangement of pseudolines, G is scale embedded in its dual, and any edge of G is contained in the cut-set of a convex cut of G.