398 research outputs found
On powers of interval graphs and their orders
It was proved by Raychaudhuri in 1987 that if a graph power is an
interval graph, then so is the next power . This result was extended to
-trapezoid graphs by Flotow in 1995. We extend the statement for interval
graphs by showing that any interval representation of can be extended
to an interval representation of that induces the same left endpoint and
right endpoint orders. The same holds for unit interval graphs. We also show
that a similar fact does not hold for trapezoid graphs.Comment: 4 pages, 1 figure. It has come to our attention that Theorem 1, the
main result of this note, follows from earlier results of [G. Agnarsson, P.
Damaschke and M. M. Halldorsson. Powers of geometric intersection graphs and
dispersion algorithms. Discrete Applied Mathematics 132(1-3):3-16, 2003].
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Density theorems for intersection graphs of t-monotone curves
A curve \gamma in the plane is t-monotone if its interior has at most t-1
vertical tangent points. A family of t-monotone curves F is \emph{simple} if
any two members intersect at most once. It is shown that if F is a simple
family of n t-monotone curves with at least \epsilon n^2 intersecting pairs
(disjoint pairs), then there exists two subfamilies F_1,F_2 \subset F of size
\delta n each, such that every curve in F_1 intersects (is disjoint to) every
curve in F_2, where \delta depends only on \epsilon. We apply these results to
find pairwise disjoint edges in simple topological graphs
Graph isomorphism completeness for trapezoid graphs
The complexity of the graph isomorphism problem for trapezoid graphs has been
open over a decade. This paper shows that the problem is GI-complete. More
precisely, we show that the graph isomorphism problem is GI-complete for
comparability graphs of partially ordered sets with interval dimension 2 and
height 3. In contrast, the problem is known to be solvable in polynomial time
for comparability graphs of partially ordered sets with interval dimension at
most 2 and height at most 2.Comment: 4 pages, 3 Postscript figure
On the intersection of tolerance and cocomparability graphs.
It has been conjectured by Golumbic and Monma in 1984 that the intersection of tolerance and cocomparability graphs coincides with bounded tolerance graphs. Since cocomparability graphs can be efficiently recognized, a positive answer to this conjecture in the general case would enable us to efficiently distinguish between tolerance and bounded tolerance graphs, although it is NP-complete to recognize each of these classes of graphs separately. The conjecture has been proved under some – rather strong – structural assumptions on the input graph; in particular, it has been proved for complements of trees, and later extended to complements of bipartite graphs, and these are the only known results so far. Furthermore, it is known that the intersection of tolerance and cocomparability graphs is contained in the class of trapezoid graphs. In this article we prove that the above conjecture is true for every graph G, whose tolerance representation satisfies a slight assumption; note here that this assumption concerns only the given tolerance representation R of G, rather than any structural property of G. This assumption on the representation is guaranteed by a wide variety of graph classes; for example, our results immediately imply the correctness of the conjecture for complements of triangle-free graphs (which also implies the above-mentioned correctness for complements of bipartite graphs). Our proofs are algorithmic, in the sense that, given a tolerance representation R of a graph G, we describe an algorithm to transform R into a bounded tolerance representation R  ∗  of G. Furthermore, we conjecture that any minimal tolerance graph G that is not a bounded tolerance graph, has a tolerance representation with exactly one unbounded vertex. Our results imply the non-trivial result that, in order to prove the conjecture of Golumbic and Monma, it suffices to prove our conjecture. In addition, there already exists evidence in the literature that our conjecture is true
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