Finding All Convex Cuts of a Plane Graph in Cubic Time

In this paper we address the task of finding convex cuts of a graph. In addition to the theoretical value of drawing a connection between geometric and combinatorial objects, cuts with this or related properties can be beneficial in various applications, e. g., routing in road networks and mesh partitioning. It is known that the decision problem whether a general graph is k-convex is NP-complete for fixed k ≥ 2. However, we show that for plane graphs all convex cuts (i. e., k = 2) can be computed in polynomial time. To this end we first restrict our consideration to a subset of plane `1-graphs for which the so-called alternating cuts can be embedded as plane curves such that the plane curves form an arrangement of pseudolines. For a graph G in this set we formulate a one-to-one correspondence between the plane curves and the convex cuts of a bipartite graph from which G can be recovered. Due to their local nature, alternating cuts cannot guide the search for convex cuts in more general graphs. Therefore we modify the concept of alternating cuts using the Djoković relation, which is of global nature and gives rise to cuts of bipartite graphs. We first present an algorithm that computes all convex cuts of a (not necessarily plane) bipartite graph H ′ = (V,E) in O(|E|) time. Then we establish a connection between convex cuts of a graph H and the Djoković relation on a (bipartite) subdivision H ′ of H. Finally, we use this connection to compute all convex cuts of a plane graph in cubic time.