1,295 research outputs found
Finding all Convex Cuts of a Plane Graph in Polynomial Time
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 is a
-partition such that both and are convex,
\ie shortest paths between vertices in never leave , . Finding convex cuts is -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 is bipartite and the Djokovic
relation is transitive on , \ie is a partial cube, then the cut-sets of
's convex cuts are precisely the equivalence classes of the Djokovic
relation. In particular, any edge of 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
using two binary relations on edges: (i) the Djokovic relation on the edges of
a subdivision of , where any edge of is subdivided into exactly two
edges and (ii) a relation on the edges of 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
Convexity in partial cubes: the hull number
We prove that the combinatorial optimization problem of determining the hull
number of a partial cube is NP-complete. This makes partial cubes the minimal
graph class for which NP-completeness of this problem is known and improves
some earlier results in the literature.
On the other hand we provide a polynomial-time algorithm to determine the
hull number of planar partial cube quadrangulations.
Instances of the hull number problem for partial cubes described include
poset dimension and hitting sets for interiors of curves in the plane.
To obtain the above results, we investigate convexity in partial cubes and
characterize these graphs in terms of their lattice of convex subgraphs,
improving a theorem of Handa. Furthermore we provide a topological
representation theorem for planar partial cubes, generalizing a result of
Fukuda and Handa about rank three oriented matroids.Comment: 19 pages, 4 figure
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
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