22,509 research outputs found
Line-distortion, Bandwidth and Path-length of a graph
We investigate the minimum line-distortion and the minimum bandwidth problems
on unweighted graphs and their relations with the minimum length of a
Robertson-Seymour's path-decomposition. The length of a path-decomposition of a
graph is the largest diameter of a bag in the decomposition. The path-length of
a graph is the minimum length over all its path-decompositions. In particular,
we show:
- if a graph can be embedded into the line with distortion , then
admits a Robertson-Seymour's path-decomposition with bags of diameter at most
in ;
- for every class of graphs with path-length bounded by a constant, there
exist an efficient constant-factor approximation algorithm for the minimum
line-distortion problem and an efficient constant-factor approximation
algorithm for the minimum bandwidth problem;
- there is an efficient 2-approximation algorithm for computing the
path-length of an arbitrary graph;
- AT-free graphs and some intersection families of graphs have path-length at
most 2;
- for AT-free graphs, there exist a linear time 8-approximation algorithm for
the minimum line-distortion problem and a linear time 4-approximation algorithm
for the minimum bandwidth problem
Even-cycle decompositions of graphs with no odd--minor
An even-cycle decomposition of a graph G is a partition of E(G) into cycles
of even length. Evidently, every Eulerian bipartite graph has an even-cycle
decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian
planar graph with an even number of edges also admits an even-cycle
decomposition. Later, Zhang (1994) generalized this to graphs with no
-minor.
Our main theorem gives sufficient conditions for the existence of even-cycle
decompositions of graphs in the absence of odd minors. Namely, we prove that
every 2-connected loopless Eulerian odd--minor-free graph with an even
number of edges has an even-cycle decomposition.
This is best possible in the sense that `odd--minor-free' cannot be
replaced with `odd--minor-free.' The main technical ingredient is a
structural characterization of the class of odd--minor-free graphs, which
is due to Lov\'asz, Seymour, Schrijver, and Truemper.Comment: 17 pages, 6 figures; minor revisio
A note on bounds for the cop number using tree decompositions
In this short note, we supply a new upper bound on the cop number in terms of
tree decompositions. Our results in some cases extend a previously derived
bound on the cop number using treewidth
A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface
Given a graph cellularly embedded on a surface of genus , a
cut graph is a subgraph of such that cutting along yields a
topological disk. We provide a fixed parameter tractable approximation scheme
for the problem of computing the shortest cut graph, that is, for any
, we show how to compute a approximation of
the shortest cut graph in time .
Our techniques first rely on the computation of a spanner for the problem
using the technique of brick decompositions, to reduce the problem to the case
of bounded tree-width. Then, to solve the bounded tree-width case, we introduce
a variant of the surface-cut decomposition of Ru\'e, Sau and Thilikos, which
may be of independent interest
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