564 research outputs found
Applications of a new separator theorem for string graphs
An intersection graph of curves in the plane is called a string graph.
Matousek almost completely settled a conjecture of the authors by showing that
every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log
m). In the present note, this bound is combined with a result of the authors,
according to which every dense string graph contains a large complete balanced
bipartite graph. Three applications are given concerning string graphs G with n
vertices: (i) if K_t is not a subgraph of G for some t, then the chromatic
number of G is at most (\log n)^{O(\log t)}; (ii) if K_{t,t} is not a subgraph
of G, then G has at most t(\log t)^{O(1)}n edges,; and (iii) a lopsided
Ramsey-type result, which shows that the Erdos-Hajnal conjecture almost holds
for string graphs.Comment: 7 page
Dyck path triangulations and extendability
We introduce the Dyck path triangulation of the cartesian product of two
simplices . The maximal simplices of this
triangulation are given by Dyck paths, and its construction naturally
generalizes to produce triangulations of
using rational Dyck paths. Our study of the Dyck path triangulation is
motivated by extendability problems of partial triangulations of products of
two simplices. We show that whenever , any triangulation of
extends to a unique triangulation of
. Moreover, with an explicit construction, we
prove that the bound is optimal. We also exhibit interesting
interpretations of our results in the language of tropical oriented matroids,
which are analogous to classical results in oriented matroid theory.Comment: 15 pages, 14 figures. Comments very welcome
The Effect of Planarization on Width
We study the effects of planarization (the construction of a planar diagram
from a non-planar graph by replacing each crossing by a new vertex) on
graph width parameters. We show that for treewidth, pathwidth, branchwidth,
clique-width, and tree-depth there exists a family of -vertex graphs with
bounded parameter value, all of whose planarizations have parameter value
. However, for bandwidth, cutwidth, and carving width, every graph
with bounded parameter value has a planarization of linear size whose parameter
value remains bounded. The same is true for the treewidth, pathwidth, and
branchwidth of graphs of bounded degree.Comment: 15 pages, 6 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Partitioning Perfect Graphs into Stars
The partition of graphs into "nice" subgraphs is a central algorithmic
problem with strong ties to matching theory. We study the partitioning of
undirected graphs into same-size stars, a problem known to be NP-complete even
for the case of stars on three vertices. We perform a thorough computational
complexity study of the problem on subclasses of perfect graphs and identify
several polynomial-time solvable cases, for example, on interval graphs and
bipartite permutation graphs, and also NP-complete cases, for example, on grid
graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor
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