Finding all Convex Cuts of a Plane Graph in Polynomial Time

Convexity is a notion that has been defined for subsets of R and for subsets of general graphs. A convex cut of a graph G = (V,E) is a 2-partition V1∪̇V2 = V such that both V1 and V2 are convex, i. e., shortest paths between vertices in Vi never leave Vi, i ∈ {1, 2}. Finding convex cuts is NP-hard for general graphs. To characterize convex cuts, we employ the Djoković relation, a reflexive and symmetric relation on the edges of a graph that is based on shortest paths between the edges’ end vertices. It is known for a long time that, if G is bipartite and the Djoković relation is transitive on G, i. e., G is a partial cube, then the cut-sets of G’s convex cuts are precisely the equivalence classes of the Djoković relation. In particular, any edge of G is contained in the cut-set of exactly one convex cut. We first characterize a class of plane graphs that we call well-arranged. These graphs are not necessarily partial cubes, but any edge of a well-arranged graph is contained in the cut-set(s) of at least one convex cut. Moreover, the cuts can be embedded into the plane such that they form an arrangement of pseudolines, or a slight generalization thereof. Although a well-arranged graph G is not necessarily a partial cube, there always exists a partial cube that contains a subdivision of G. We also present an algorithm that uses the Djoković relation for computing all convex cuts of a (not necessarily plane) bipartite graph in O(|E|) time. Specifically, a cut-set is the cut-set of a convex cut if and only if the Djoković relation holds for any pair of edges in the cut-set. We then characterize the cut-sets of the convex cuts of a general graph H using two binary relations on edges: (i) the Djoković relation on the edges of a subdivision of H , where any edge of H is subdivided into exactly two edges and (ii) a relation on the edges of H itself that is not the Djoković relation. Finally, we use this characterization to present the first algorithm for finding all convex cuts of a plane graph in polynomial time.