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. The conjecture has been proved under some - rather strong - \emph{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. 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 here that this assumption concerns only the given tolerance \emph{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 $a,b,c\in V$ such that $N(a) \subset N(b) \subset N(c)$; 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 $R^{\ast}$ of $G$. Furthermore, we conjecture that any \emph{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.


Introduction
and a set t = {t u | u ∈ V } of positive numbers, such that for any two vertices u, v ∈ V , uv ∈ E if and only if |I u ∩ I v | ≥ min{t u , t v }. The pair I, t is called a tolerance representation of G. A vertex u of G is called a bounded vertex (in a certain tolerance representation I, t of G) if t u ≤ |I u |; otherwise, u is called an unbounded vertex of G. If G has a tolerance representation I, t where all vertices are bounded, then G is called a bounded tolerance graph and I, t a bounded tolerance representation of G.
Given an undirected graph G = (V, E) and a vertex subset M ⊆ V , M is called a module in G, if for every u, v ∈ M and every x ∈ V \M , x is either adjacent in G to both u and v or to none of them. Note that ∅, V , and all singletons {v}, where v ∈ V , are trivial modules in G. A comparability graph is a graph which can be transitively oriented. A cocomparability graph is a graph whose complement is a comparability graph. A trapezoid (resp. parallelogram and permutation) graph is the intersection graph of trapezoids (resp. parallelograms and line segments) between two parallel lines L 1 and L 2 [8]. Such a representation with trapezoids (resp. parallelograms and line segments) is called a trapezoid (resp. parallelogram and permutation) representation of this graph. A graph is bounded tolerance if and only if it is a parallelogram graph [2]. Permutation graphs are a strict subset of parallelogram graphs [3]. Furthermore, parallelogram graphs are a strict subset of trapezoid graphs [19], and both are subsets of cocomparability graphs [8,12]. On the other hand, not every tolerance graph is a cocomparability graph [8,12].
Cocomparability graphs have received considerable attention in the literature, mainly due to their interesting structure that leads to efficient algorithms for several NP-hard problems, see e.g. [5,6,12,14]. Furthermore, the intersection of the class of cocomparability graphs with other graph classes has interesting properties and coincides with other widely known graph classes. For instance, their intersection with chordal graphs is the class of interval graphs, while their intersection with comparability graphs is the class of permutation graphs [8]. These structural characterizations find also direct algorithmic implications to the recognition problem of interval and permutation graphs, respectively, since the class of cocomparability graphs can be recognized efficiently [8,20]. In this context, the following conjecture has been made in 1984 [10]: Conjecture 1 ( [10]). The intersection of cocomparability graphs with tolerance graphs is exactly the class of bounded tolerance graphs.
Note that the inclusion in one direction is immediate: every bounded tolerance graph is a cocomparability graph [8,12], as well as a tolerance graph by definition. Conjecture 1 has been proved for complements of trees [1], and later extended to complements of bipartite graphs [18], and these are the only known results so far. Furthermore, it has been proved that the intersection of tolerance and cocomparability graphs is contained in the class of trapezoid graphs [7].
Since cocomparability graphs can be efficiently recognized [20], a positive answer to Conjecture 1 would enable us to efficiently distinguish between tolerance and bounded tolerance graphs, although it is NP-complete to recognize both tolerance and bounded tolerance graphs [16]. Recently, an intersection model for general tolerance graphs has been presented in [15], given by 3D-parallelepipeds. This parallelepiped representation of tolerance graphs generalizes the parallelogram representation of bounded tolerance graphs; the main idea is to exploit the third dimension to capture the information given by unbounded tolerances.
Our contribution. In this article we prove that Conjecture 1 is true for every graph G, whose parallelepiped representation R satisfies a slight assumption (to be defined later). This assumption is guaranteed by a wide variety of graph classes; for example, our results immediately imply correctness of the conjecture for complements of triangle-free graphs (which also implies the above mentioned correctness for complements of trees [1] and complements of bipartite graphs [18]). Furthermore, we state a new conjecture regarding only the separating examples between tolerance and bounded tolerance graphs (cf. Conjecture 2). There already exists evidence in the literature that this conjecture is true [12]. Our results reduce Conjecture 1 to our conjecture; that is, the correctness of our conjecture implies the correctness of Conjecture 1.
Specifically, we state three conditions on the unbounded vertices of G (in the parallelepiped representation R). Condition 1 is that R has exactly one unbounded vertex. Condition 2 is that, for every unbounded vertex u of G (in R), there exists no unbounded vertex v of G whose neighborhood is strictly included in the neighborhood of u. Note that these two conditions concern only the parallelepiped representation R; furthermore, the second condition is weaker than the first one. Then, Condition 3 concerns also the position of the unbounded vertices in the trapezoid representation R T , and it is weaker than the other two.
Assuming that this (weaker) Condition 3 holds, we algorithmically construct a parallelogram representation of G, thus proving that G is a bounded tolerance graph. The proof of correctness relies on the fact that G can be represented simultaneously by R and by R T . The main idea is to iteratively "eliminate" the unbounded vertices of R. That is, assuming that the input representation R has k ≥ 1 unbounded vertices, we choose an unbounded vertex u in R and construct a parallelepiped representation R * of G with k −1 unbounded vertices; specifically, R * has the same unbounded vertices as R except for u (which becomes bounded in R * ). As a milestone in the above construction of the representation R * , we construct an induced subgraph G 0 of G that includes u, with the property that the vertex set of G 0 \ {u} is a module in G \ {u}. The presented techniques are new and provide geometrical insight for the graphs that are both tolerance and cocomparability.
In order to state our Conjecture 2, we define a graph G to be a minimally unbounded tolerance graph, if G is tolerance but not bounded tolerance, while G becomes bounded tolerance if we remove any vertex of G.

Conjecture 2.
Any minimally unbounded tolerance graph has a tolerance representation with exactly one unbounded vertex.
In other words, Conjecture 2 states that any minimally unbounded tolerance graph G has a tolerance representation (or equivalently, a parallelepiped representation) R that satisfies Condition 1 (stated above). Our results imply the non-trivial result that, in order to prove Conjecture 1, it suffices to prove Conjecture 2. In addition, there already exists evidence that Conjecture 2 is true, as to the best of our knowledge it is true for all known examples of minimally unbounded tolerance graphs in the literature (see e.g. [12]).
Organization of the paper. We first review in Section 2 some properties of tolerance and trapezoid graphs. Then we define the notion of a projection representation of a tolerance graph G, which is an alternative way to think about a parallelepiped representation of G. Furthermore, we introduce the right and left border properties of a vertex in a projection representation, which are crucial for our analysis. In Section 3 we prove our main results and we discuss how these results reduce Conjecture 1 to Conjecture 2. Finally, we discuss the presented results and further research in Section 4. Due to space limitations, the proofs are omitted here; a full version of the paper can be found in [17].

Definitions and Basic Properties
Notation. We consider in this article simple undirected graphs with no loops or multiple edges. In a graph G = (V, E), the edge between vertices u and v is denoted by uv, and in this case u and v are called adjacent in G. Given a vertex subset S ⊆ V , G[S] denotes the induced subgraph of G on the vertices in S. Whenever it is clear from the context, we may not distinguish between a vertex set S and the induced subgraph G[S] of G. Furthermore, we denote for simplicity Consider a trapezoid graph G = (V, E) and a trapezoid representation R T of G, where vertex u ∈ V corresponds to the trapezoid T u in R T . Since trapezoid graphs are also cocomparability graphs [8], we can define the partial order (V, RT ), such that u RT v, or equivalently T u RT T v , if and only if T u lies completely to the left of T v in R T (and thus also uv / ∈ E). Note that there are several trapezoid representations of a particular trapezoid graph G. Given one such representation R T , we can obtain another one R T by vertical axis flipping of R T , i.e. R T is the mirror image of R T along an imaginary line perpendicular to L 1 and L 2 .
Let us now briefly review the parallelepiped representation model of tolerance graphs [15]. Consider a tolerance graph G = (V, E) and let V B and V U denote the set of bounded and unbounded vertices of G (for a certain tolerance representation), respectively. Consider now two parallel lines L 1 and L 2 in the plane. For every vertex u ∈ V , consider a parallelogram P u with two of its lines on L 1 and L 2 , respectively, and φ u be the (common) slope of the other two lines of P u with L 1 and L 2 . For every unbounded vertex u ∈ V U , the parallelogram P u is trivial, i.e. a line. In the model of [15], every bounded vertex u ∈ V B corresponds to the parallelepiped P u = {(x, y, z) | (x, y) ∈ P u , 0 ≤ z ≤ φ u } in the 3-dimensional space, while every unbounded vertex u ∈ V U corresponds to the line P u = {(x, y, z) | (x, y) ∈ P u , z = φ u }. The resulting set {P u | u ∈ V } of parallelepipeds in the 3-dimensional space constitutes the parallelepiped representation of G. In this model, two vertices u, v are adjacent if and only if P u ∩ P v = ∅. That is, R is an intersection model for G. For more details we refer to [15]. An example of a parallelepiped representation R is illustrated in Figure 1(a). This representation corresponds to the induced path P 4 = (z, u, v, w) with four vertices (P 4 is a tolerance graph); in particular, vertex w is unbounded in R, while the vertices z, u, v are bounded in R.

Definition 2 ([15]). A parallelepiped representation R of a tolerance graph G is called canonical if every unbounded vertex in R is inevitable.
For example, the parallelepiped representation of Figure 1(a) is canonical, since w is the only unbounded vertex and it is inevitable. A canonical representation of a tolerance graph G always exists, and can be computed in O(n log n) time, given a parallelepiped representation of G, where n is the number of vertices of G [15].
Given a parallelepiped representation R of the tolerance graph G, we define now an alternative representation, as follows. Let P u be the projection of P u to the plane z = 0 for every u ∈ V . Then, for two bounded vertices u and v, uv ∈ E if and only if P u ∩ P v = ∅. Furthermore, for a bounded vertex v and an unbounded vertex u, uv ∈ E if and only if P u ∩ P v = ∅ and φ v > φ u . Moreover, two unbounded vertices u and v of G are never adjacent (even in the case where P u intersects P v ). In the following, we will call such a representation a projection representation of a tolerance graph. Note that P u is a parallelogram (resp. a line segment) if u is bounded (resp. unbounded). The projection representation that corresponds to the parallelepiped representation of Figure 1(a) is presented in Figure 1(b). In the sequel, given a tolerance graph G, we will call a projection representation R of G a canonical representation of G, if R is the projection representation that is implied by a canonical parallelepiped representation of G.
In the example of Figure 1, the projection representation R is canonical, since the parallelepiped representation R is canonical as well.
Let R be a projection representation of a tolerance graph G = (V, E). For every parallelogram P u in R, where u ∈ V , we define by l(u) and r(u) (resp. L(u) and R(u)) the lower (resp. upper) left and right endpoint of P u , respectively (cf. the parallelogram P v in Figure 1(b)). Note that l(u) = r(u) and L(u) = R(u) for every unbounded vertex u. We assume throughout the paper w.l.o.g. that all endpoints and all slopes of the parallelograms in a projection representation are distinct [12,13,15]. For simplicity of the presentation, we will denote in the following P u just by P u in any projection representation.
Similarly to a trapezoid representation, we can define the relation R also for a projection representation R. Namely, P u R P v if and only if P u lies completely to the left of P v in R. Otherwise, if neither P u R P v nor P v R P u , we will say that P u intersects P v in R, i.e. P u ∩ P v = ∅ in R. Note that, for two vertices u and v of a tolerance graph G = (V, E), P u may intersect P v in a projection representation R of G, although u is not adjacent to v in G, i.e. uv / ∈ E. Thus, a projection representation R of a tolerance graph G is not necessarily an intersection model for G.
In [11,15] the hovering set of an unbounded vertex in a tolerance graph has been defined. According to these definitions, the hovering set depends on a particular representation of the tolerance graph. In the following, we extend this definition to the notion of covering vertices of an arbitrary graph G, which is independent of any representation of G.

Definition 3. Let G = (V, E) be an arbitrary graph and u ∈ V be a vertex of
In the following, for simplicity of the presentation, we may not distinguish between the connected components of V 0 (u) and the vertex set of these components. In the next definition we introduce the notion of the right (resp. left) border property of a vertex u in a projection representation R of a tolerance graph G. This notion is of particular importance for the sequel of the paper.

Definition 4.
Let G = (V, E) be a tolerance graph, u be an arbitrary vertex of G, and R be a projection representation of G. Then, u has the right (resp. left) border property in R, if there exists no pair of vertices w ∈ N (u) and x ∈ V 0 (u), such that P w R P x (resp. P x R P w ).
We denote in the sequel by Tolerance the class of tolerance graphs, and we use the corresponding notations for the classes of bounded tolerance, cocomparability, and trapezoid graphs. Let G ∈ Tolerance ∩ Cocomparability. Then G is also a trapezoid graph [7]. Thus, since Trapezoid ⊆ Cocomparability, it follows that Tolerance ∩ Cocomparability = Tolerance ∩ Trapezoid. Furthermore, clearly Bounded Tolerance ⊆ (Tolerance ∩ Trapezoid), since Bounded Tolerance ⊆ Tolerance and Bounded Tolerance ⊆ Trapezoid. In the following we consider a graph G ∈ (Tolerance ∩ Trapezoid) \ Bounded Tolerance, assuming that one exists, and our aim is to get to a contradiction; namely, to prove that (Tolerance ∩ Trapezoid) = Bounded Tolerance.

Main Results
In this section, we prove that for a graph G ∈ (Tolerance ∩ Trapezoid) it follows that also G ∈ Bounded Tolerance by making a slight assumption on the unbounded vertices of a projection representation R of G. In particular, we choose a certain unbounded vertex u in R and we "eliminate" u in R in the following sense: assuming that R has k ≥ 1 unbounded vertices, we construct a projection representation R * of G with k − 1 unbounded vertices, where all bounded vertices remain bounded and u is transformed to a bounded vertex. In Section 3.1 we deal with the case where the unbounded vertex u has the right or the left border property in R, while in Section 3.2 we deal with the case where u has neither the left nor the right border property in R. Finally we combine these two results in Section 3.3, in order to eliminate all k unbounded vertices in R, regardless of whether or not they have the right or left border property.

The Case Where u Has the Right or the Left Border Property
In this section we consider an arbitrary unbounded vertex u of G in the projection representation R, and we assume that u has the right or the left border property in R. Then, as we prove in the next theorem, there exists another projection representation R * of G, in which u has been replaced by a bounded vertex.
Tolerance with the smallest number of vertices. Let R be a projection representation of G with k unbounded vertices and u be an unbounded vertex in R. If u has the right or the left border property in R, then G has a projection representation R * with k − 1 unbounded vertices.

The Case Where u Has Neither the Left Nor the Right Border Property
In this section we consider a graph G ∈ (Tolerance ∩ Trapezoid) \ Bounded Tolerance with the smallest number of vertices. Furthermore, let R and R T be a canonical projection and a trapezoid representation of G, respectively, and u be an unbounded vertex of G in R. We consider the case where G has no unbounded vertex in R with the right or the left border property (otherwise Theorem 1 can be applied). It can be proved that V 0 (u) = ∅ and that V 0 (u) is connected (cf. [17]). Therefore, since u is not adjacent to any vertex of V 0 (u) by Definition 3, either all trapezoids of V 0 (u) lie to the left, or all to the right of T u in R T . Consider first the case where all trapezoids of V 0 (u) lie to the left of T u in R T , i.e. T x RT T u for every x ∈ V 0 (u). It can be proved that N (v) = N (u) for every unbounded vertex v = u in R (cf. [17]). Denote by Q u = {v ∈ V U | N (v) ⊂ N (u)} the set of unbounded vertices v of G in R, whose neighborhood set is included in the neighborhood set of u. Since no two unbounded vertices are adjacent, we can partition the set Q u into the two subsets Furthermore, it can be proved that T v RT T x RT T u for every v ∈ Q 1 (u) and every x ∈ V 0 (u) (cf. [17]). That is, Consider now the case where all trapezoids of V 0 (u) lie to the right of T u in R T , i.e. T u RT T x for every x ∈ V 0 (u). Then, by performing vertical axis flipping of R T , we partition similarly to the above the set Q u into the sets Q 1 (u) and Q 2 (u). That is, in this (symmetric) case, the sets Q 1 (u) and Q 2 (u) will be We state now three conditions on G, regarding the unbounded vertices in R; the third one depends also on the representation R T . Note that the second condition is weaker than the first one, while the third one is weaker than the other two. Then, we prove Theorem 2, assuming that the third condition holds. First, we introduce the notion of neighborhood maximality for unbounded vertices in a tolerance graph.

Definition 5.
Let G be a tolerance graph, R be a projection representation of G, and u be an unbounded vertex in R. Then, u is unbounded-maximal if there is no unbounded vertex v in R, such that N (u) ⊂ N (v).

Condition 1.
The projection representation R of G has exactly one unbounded vertex.

Condition 2.
For every unbounded vertex u of G in R, Q u = ∅; namely, all unbounded vertices are unbounded-maximal.

Condition 3. For every unbounded vertex
In the following of the section we assume that Condition 3 holds, which is weaker than Conditions 1 and 2. We present now the main theorem of this section. The proof of this theorem is based on the fact that G has simultaneously the two representations R and R T .
Tolerance with the smallest number of vertices. Let R T be a trapezoid representation of G and R be a projection representation of G with k unbounded vertices. Then, assuming that Condition 3 holds, there exists a projection representation R * of G with k − 1 unbounded vertices.

The General Case
Recall now that Tolerance ∩ Cocomparability = Tolerance ∩ Trapezoid [7]. The next main theorem follows by recursive application of Theorem 2.
Theorem 3. Let G = (V, E) ∈ (Tolerance ∩ Cocomparability), R T be a trapezoid representation of G, and R be a projection representation of G. Then, assuming that one of the Conditions 1, 2, or 3 holds, G is a bounded tolerance graph.
As an immediate implication of Theorem 3, we prove in the next corollary that Conjecture 1 is true in particular for every graph G that has no three independent vertices a, b, c such that N (a) ⊂ N (b) ⊂ N (c), since Condition 2 is guaranteed to be true for every such graph G. Therefore, in particular, the conjecture is also true for the complements of triangle-free graphs.  N (c). Then, G is also a bounded tolerance graph. Definition 6. Let G ∈ Tolerance \ Bounded Tolerance. If G \ {u} is a bounded tolerance graph for every vertex of G, then G is a minimally unbounded tolerance graph.
Assume now that Conjecture 1 is not true, and let G be a counterexample with the smallest number of vertices. Then, in particular, G is a minimally unbounded tolerance graph by Definition 6. Now, if our Conjecture 2 is true (see Section 1), then G has a projection representation R with exactly one unbounded vertex, i.e. R satisfies Condition 1. Thus, G is a bounded tolerance graph by Theorem 3, which is a contradiction, since G has been assumed to be a counterexample to Conjecture 1. Thus, we obtain the following theorem. Therefore, in order to prove Conjecture 1, it suffices to prove Conjecture 2. In addition, there already exists evidence that this conjecture is true, as to the best of our knowledge all known examples of minimally unbounded tolerance graphs have a tolerance representation with exactly one unbounded vertex; for such examples, see e.g. [12].

Concluding Remarks and Open Problems
In this article we dealt with the over 25 years old conjecture of [10], which states that if a graph G is both tolerance and cocomparability, then it is also bounded tolerance. Specifically, we proved that the conjecture is true for every graph G, whose tolerance representation R of G satisfies a slight assumption, instead of making any structural assumption on G -as it was the case in all previously known results. Furthermore, we conjectured that any minimal graph G that is a tolerance but 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 [10], it suffices to prove our conjecture. In addition, there already exists evidence in the literature that this conjecture is true [12].