Convexity is a notion that has been defined for subsets of \RR^n 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,
\ie 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 Djokovic 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 Djokovic
relation is transitive on G, \ie G is a partial cube, then the cut-sets of
G's convex cuts are precisely the equivalence classes of the Djokovic
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 {\em
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. We also present an algorithm that uses the Djokovic relation for computing
all convex cuts of a (not necessarily plane) bipartite graph in \bigO(|E|^3)
time. Specifically, a cut-set is the cut-set of a convex cut if and only if the
Djokovic 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 Djokovic 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 Djokovic
relation. Finally, we use this characterization to present the first algorithm
for finding all convex cuts of a plane graph in polynomial time.Comment: 23 pages. Submitted to Journal of Discrete Algorithms (JDA