12,736 research outputs found
Fast calculation of the variance of edge crossings
The crossing number, i.e. the minimum number of edge crossings arising when
drawing a graph on a certain surface, is a very important problem of graph
theory. The opposite problem, i.e. the maximum crossing number, is receiving
growing attention. Here we consider a complementary problem of the distribution
of the number of edge crossings, namely the variance of the number of
crossings, when embedding the vertices of an arbitrary graph in some space at
random. In his pioneering research, Moon derived that variance on random linear
arrangements of complete unipartite and bipartite graphs. Given the need of
efficient algorithms to support this sort of research and given also the
growing interest of the number of edge crossings in spatial networks, networks
where vertices are embedded in some space, here we derive algorithms to
calculate the variance in arbitrary graphs in -time, and in forests in
-time. These algorithms work on a wide range of random layouts (not only
on Moon's) and are based on novel arithmetic expressions for the calculation of
the variance that we develop from previous theoretical work. This paves the way
for many applications that rely on a fast but exact calculation of the
variance.Comment: Better connection with graph theory (crossing number). Introduction
and discussion substantially rewritten. Minor corrections in other parts of
the articl
Dynamic Programming for Graphs on Surfaces
We provide a framework for the design and analysis of dynamic programming
algorithms for surface-embedded graphs on n vertices and branchwidth at most k.
Our technique applies to general families of problems where standard dynamic
programming runs in 2^{O(k log k)} n steps. Our approach combines tools from
topological graph theory and analytic combinatorics. In particular, we
introduce a new type of branch decomposition called "surface cut
decomposition", generalizing sphere cut decompositions of planar graphs
introduced by Seymour and Thomas, which has nice combinatorial properties.
Namely, the number of partial solutions that can be arranged on a surface cut
decomposition can be upper-bounded by the number of non-crossing partitions on
surfaces with boundary. It follows that partial solutions can be represented by
a single-exponential (in the branchwidth k) number of configurations. This
proves that, when applied on surface cut decompositions, dynamic programming
runs in 2^{O(k)} n steps. That way, we considerably extend the class of
problems that can be solved in running times with a single-exponential
dependence on branchwidth and unify/improve most previous results in this
direction.Comment: 28 pages, 3 figure
Surface Split Decompositions and Subgraph Isomorphism in Graphs on Surfaces
The Subgraph Isomorphism problem asks, given a host graph G on n vertices and
a pattern graph P on k vertices, whether G contains a subgraph isomorphic to P.
The restriction of this problem to planar graphs has often been considered.
After a sequence of improvements, the current best algorithm for planar graphs
is a linear time algorithm by Dorn (STACS '10), with complexity .
We generalize this result, by giving an algorithm of the same complexity for
graphs that can be embedded in surfaces of bounded genus. At the same time, we
simplify the algorithm and analysis. The key to these improvements is the
introduction of surface split decompositions for bounded genus graphs, which
generalize sphere cut decompositions for planar graphs. We extend the algorithm
for the problem of counting and generating all subgraphs isomorphic to P, even
for the case where P is disconnected. This answers an open question by Eppstein
(SODA '95 / JGAA '99)
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