9,233 research outputs found
Crossing and weighted crossing number of near-planar graphs
A nonplanar graph G is near-planar if it contains an edge e such that G − e is planar. The problem of determining the crossing number of a near-planar graph is exhibited from different combinatorial viewpoints. On the one hand, we develop min-max formulas involving efficiently computable lower and upper bounds. These min-max results are the first of their kind in the study of crossing numbers and improve the approximation factor for the approximation algorithm given by Hliněny and Salazar (Graph Drawing GD’06). On the other hand, we show that it is NP-hard to compute a weighted version of the crossing number for near-planar graphs
On Hardness of the Joint Crossing Number
The Joint Crossing Number problem asks for a simultaneous embedding of two
disjoint graphs into one surface such that the number of edge crossings
(between the two graphs) is minimized. It was introduced by Negami in 2001 in
connection with diagonal flips in triangulations of surfaces, and subsequently
investigated in a general form for small-genus surfaces. We prove that all of
the commonly considered variants of this problem are NP-hard already in the
orientable surface of genus 6, by a reduction from a special variant of the
anchored crossing number problem of Cabello and Mohar
On the pseudolinear crossing number
A drawing of a graph is {\em pseudolinear} if there is a pseudoline
arrangement such that each pseudoline contains exactly one edge of the drawing.
The {\em pseudolinear crossing number} of a graph is the minimum number of
pairwise crossings of edges in a pseudolinear drawing of . We establish
several facts on the pseudolinear crossing number, including its computational
complexity and its relationship to the usual crossing number and to the
rectilinear crossing number. This investigation was motivated by open questions
and issues raised by Marcus Schaefer in his comprehensive survey of the many
variants of the crossing number of a graph.Comment: 12 page
NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs
In 1988, Vazirani gave an NC algorithm for computing the number of perfect
matchings in -minor-free graphs by building on Kasteleyn's scheme for
planar graphs, and stated that this "opens up the possibility of obtaining an
NC algorithm for finding a perfect matching in -free graphs." In this
paper, we finally settle this 30-year-old open problem. Building on recent NC
algorithms for planar and bounded-genus perfect matching by Anari and Vazirani
and later by Sankowski, we obtain NC algorithms for perfect matching in any
minor-closed graph family that forbids a one-crossing graph. This family
includes several well-studied graph families including the -minor-free
graphs and -minor-free graphs. Graphs in these families not only have
unbounded genus, but can have genus as high as . Our method applies as
well to several other problems related to perfect matching. In particular, we
obtain NC algorithms for the following problems in any family of graphs (or
networks) with a one-crossing forbidden minor:
Determining whether a given graph has a perfect matching and if so,
finding one.
Finding a minimum weight perfect matching in the graph, assuming
that the edge weights are polynomially bounded.
Finding a maximum -flow in the network, with arbitrary
capacities.
The main new idea enabling our results is the definition and use of
matching-mimicking networks, small replacement networks that behave the same,
with respect to matching problems involving a fixed set of terminals, as the
larger network they replace.Comment: 21 pages, 6 figure
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