44 research outputs found
On the intersection of tolerance and cocomparability graphs.
Tolerance graphs have been extensively studied since their introduction, due to their interesting
structure and their numerous applications, as they generalize both interval and permutation
graphs in a natural way. It has been conjectured by Golumbic, Monma, and Trotter 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. This
longstanding 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. Our main result in this article is that the above conjecture is true
for every graph G that admits a tolerance representation with exactly one unbounded vertex;
note that this assumption concerns only the given tolerance representation R of G, rather than
any structural property of G. Moreover, our results imply as a corollary that the conjecture of
Golumbic, Monma, and Trotter is true for every graph G = (V,E) that has no three independent
vertices a, b, c ∈ V such that N(a) ⊂ N(b) ⊂ N(c), where N(v) denotes the set of neighbors of
a vertex v ∈ V ; this is satisfied in particular when G is the complement of a triangle-free graph
(which also implies the above-mentioned correctness for complements of bipartite graphs). Our
proofs are constructive, in the sense that, given a tolerance representation R of a graph G,
we 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, Monma, and Trotter, it suffices to prove our conjecture
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
On-line coloring between two lines
We study on-line colorings of certain graphs given as intersection graphs of
objects "between two lines", i.e., there is a pair of horizontal lines such
that each object of the representation is a connected set contained in the
strip between the lines and touches both. Some of the graph classes admitting
such a representation are permutation graphs (segments), interval graphs
(axis-aligned rectangles), trapezoid graphs (trapezoids) and cocomparability
graphs (simple curves). We present an on-line algorithm coloring graphs given
by convex sets between two lines that uses colors on graphs with
maximum clique size .
In contrast intersection graphs of segments attached to a single line may
force any on-line coloring algorithm to use an arbitrary number of colors even
when .
The {\em left-of} relation makes the complement of intersection graphs of
objects between two lines into a poset. As an aside we discuss the relation of
the class of posets obtained from convex sets between two lines
with some other classes of posets: all -dimensional posets and all posets of
height are in but there is a -dimensional poset of height
that does not belong to .
We also show that the on-line coloring problem for curves between two lines
is as hard as the on-line chain partition problem for arbitrary posets.Comment: grant support adde