5 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 β rather strong β 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 β V such that N(a) β N(b) β N(c), where N(v) denotes the set of neighbors of
a vertex v β 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 β rather strong β 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 βββ 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 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 β rather strong β 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 βββ 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 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 β rather strong β 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 β V such that N(a) β N(b) β N(c), where N(v) denotes the set of neighbors of a vertex v β 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