6,533 research outputs found
Recognizing Weighted Disk Contact Graphs
Disk contact representations realize graphs by mapping vertices bijectively
to interior-disjoint disks in the plane such that two disks touch each other if
and only if the corresponding vertices are adjacent in the graph. Deciding
whether a vertex-weighted planar graph can be realized such that the disks'
radii coincide with the vertex weights is known to be NP-hard. In this work, we
reduce the gap between hardness and tractability by analyzing the problem for
special graph classes. We show that it remains NP-hard for outerplanar graphs
with unit weights and for stars with arbitrary weights, strengthening the
previous hardness results. On the positive side, we present constructive
linear-time recognition algorithms for caterpillars with unit weights and for
embedded stars with arbitrary weights.Comment: 24 pages, 21 figures, extended version of a paper to appear at the
International Symposium on Graph Drawing and Network Visualization (GD) 201
Unit Grid Intersection Graphs: Recognition and Properties
It has been known since 1991 that the problem of recognizing grid
intersection graphs is NP-complete. Here we use a modified argument of the
above result to show that even if we restrict to the class of unit grid
intersection graphs (UGIGs), the recognition remains hard, as well as for all
graph classes contained inbetween. The result holds even when considering only
graphs with arbitrarily large girth. Furthermore, we ask the question of
representing UGIGs on grids of minimal size. We show that the UGIGs that can be
represented in a square of side length 1+epsilon, for a positive epsilon no
greater than 1, are exactly the orthogonal ray graphs, and that there exist
families of trees that need an arbitrarily large grid
On the Recognition of Fuzzy Circular Interval Graphs
Fuzzy circular interval graphs are a generalization of proper circular arc
graphs and have been recently introduced by Chudnovsky and Seymour as a
fundamental subclass of claw-free graphs. In this paper, we provide a
polynomial-time algorithm for recognizing such graphs, and more importantly for
building a suitable representation.Comment: 12 pages, 2 figure
Mutant knots and intersection graphs
We prove that if a finite order knot invariant does not distinguish mutant
knots, then the corresponding weight system depends on the intersection graph
of a chord diagram rather than on the diagram itself. The converse statement is
easy and well known. We discuss relationship between our results and certain
Lie algebra weight systems.Comment: 13 pages, many figure
Automorphism Groups of Geometrically Represented Graphs
We describe a technique to determine the automorphism group of a
geometrically represented graph, by understanding the structure of the induced
action on all geometric representations. Using this, we characterize
automorphism groups of interval, permutation and circle graphs. We combine
techniques from group theory (products, homomorphisms, actions) with data
structures from computer science (PQ-trees, split trees, modular trees) that
encode all geometric representations.
We prove that interval graphs have the same automorphism groups as trees, and
for a given interval graph, we construct a tree with the same automorphism
group which answers a question of Hanlon [Trans. Amer. Math. Soc 272(2), 1982].
For permutation and circle graphs, we give an inductive characterization by
semidirect and wreath products. We also prove that every abstract group can be
realized by the automorphism group of a comparability graph/poset of the
dimension at most four
On the structure of (pan, even hole)-free graphs
A hole is a chordless cycle with at least four vertices. A pan is a graph
which consists of a hole and a single vertex with precisely one neighbor on the
hole. An even hole is a hole with an even number of vertices. We prove that a
(pan, even hole)-free graph can be decomposed by clique cutsets into
essentially unit circular-arc graphs. This structure theorem is the basis of
our -time certifying algorithm for recognizing (pan, even hole)-free
graphs and for our -time algorithm to optimally color them.
Using this structure theorem, we show that the tree-width of a (pan, even
hole)-free graph is at most 1.5 times the clique number minus 1, and thus the
chromatic number is at most 1.5 times the clique number.Comment: Accepted to appear in the Journal of Graph Theor
- …