1,273 research outputs found
Simultaneous Embeddability of Two Partitions
We study the simultaneous embeddability of a pair of partitions of the same
underlying set into disjoint blocks. Each element of the set is mapped to a
point in the plane and each block of either of the two partitions is mapped to
a region that contains exactly those points that belong to the elements in the
block and that is bounded by a simple closed curve. We establish three main
classes of simultaneous embeddability (weak, strong, and full embeddability)
that differ by increasingly strict well-formedness conditions on how different
block regions are allowed to intersect. We show that these simultaneous
embeddability classes are closely related to different planarity concepts of
hypergraphs. For each embeddability class we give a full characterization. We
show that (i) every pair of partitions has a weak simultaneous embedding, (ii)
it is NP-complete to decide the existence of a strong simultaneous embedding,
and (iii) the existence of a full simultaneous embedding can be tested in
linear time.Comment: 17 pages, 7 figures, extended version of a paper to appear at GD 201
A note on the Cops & Robber game on graphs embedded in non-orientable surfaces
The Cops and Robber game is played on undirected finite graphs. A number of
cops and one robber are positioned on vertices and take turns in sliding along
edges. The cops win if they can catch the robber. The minimum number of cops
needed to win on a graph is called its cop number. It is known that the cop
number of a graph embedded on a surface of genus is at most ,
if is orientable (Schroeder 2004), and at most , otherwise
(Nowakowski & Schroeder 1997).
We improve the bounds for non-orientable surfaces by reduction to the
orientable case using covering spaces.
As corollaries, using Schroeder's results, we obtain the following: the
maximum cop number of graphs embeddable in the projective plane is 3; the cop
number of graphs embeddable in the Klein Bottle is at most 4, and an upper
bound is for all other .Comment: 5 pages, 1 figur
On discrete integrable equations with convex variational principles
We investigate the variational structure of discrete Laplace-type equations
that are motivated by discrete integrable quad-equations. In particular, we
explain why the reality conditions we consider should be all that are
reasonable, and we derive sufficient conditions (that are often necessary) on
the labeling of the edges under which the corresponding generalized discrete
action functional is convex. Convexity is an essential tool to discuss
existence and uniqueness of solutions to Dirichlet boundary value problems.
Furthermore, we study which combinatorial data allow convex action functionals
of discrete Laplace-type equations that are actually induced by discrete
integrable quad-equations, and we present how the equations and functionals
corresponding to (Q3) are related to circle patterns.Comment: 39 pages, 8 figures. Revision of the whole manuscript, reorder of
sections. Major changes due to additional reality conditions for (Q3) and
(Q4): new Section 2.3; Theorem 1 and Sections 3.5, 3.6, 3.7 update
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