26,982 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
Structural parameterizations for boxicity
The boxicity of a graph is the least integer such that has an
intersection model of axis-aligned -dimensional boxes. Boxicity, the problem
of deciding whether a given graph has boxicity at most , is NP-complete
for every fixed . We show that boxicity is fixed-parameter tractable
when parameterized by the cluster vertex deletion number of the input graph.
This generalizes the result of Adiga et al., that boxicity is fixed-parameter
tractable in the vertex cover number.
Moreover, we show that boxicity admits an additive -approximation when
parameterized by the pathwidth of the input graph.
Finally, we provide evidence in favor of a conjecture of Adiga et al. that
boxicity remains NP-complete when parameterized by the treewidth.Comment: 19 page
Bandwidth theorem for random graphs
A graph is said to have \textit{bandwidth} at most , if there exists a
labeling of the vertices by , so that whenever
is an edge of . Recently, B\"{o}ttcher, Schacht, and Taraz
verified a conjecture of Bollob\'{a}s and Koml\'{o}s which says that for every
positive , there exists such that if is an
-vertex -chromatic graph with maximum degree at most which has
bandwidth at most , then any graph on vertices with minimum
degree at least contains a copy of for large enough
. In this paper, we extend this theorem to dense random graphs. For
bipartite , this answers an open question of B\"{o}ttcher, Kohayakawa, and
Taraz. It appears that for non-bipartite the direct extension is not
possible, and one needs in addition that some vertices of have independent
neighborhoods. We also obtain an asymptotically tight bound for the maximum
number of vertex disjoint copies of a fixed -chromatic graph which one
can find in a spanning subgraph of with minimum degree .Comment: 29 pages, 3 figure
Bandwidth and density for block graphs
The bandwidth of a graph G is the minimum of the maximum difference between
adjacent labels when the vertices have distinct integer labels. We provide a
polynomial algorithm to produce an optimal bandwidth labeling for graphs in a
special class of block graphs (graphs in which every block is a clique), namely
those where deleting the vertices of degree one produces a path of cliques. The
result is best possible in various ways. Furthermore, for two classes of graphs
that are ``almost'' caterpillars, the bandwidth problem is NP-complete.Comment: 14 pages, 9 included figures. Note: figures did not appear in
original upload; resubmission corrects thi
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