## 1 Introduction

“The first genuine monograph on graph theory (König, 1936) had the following subtitle: Combinatorial Topology of Systems of Segments [5]. Although graph theory and topology stem from the same root, the connection between them has somewhat faded away in the past few decades. In the most prolific new areas of graph theory including Ramsey theory, extremal graph theory and random graphs, graphs are regarded as abstract binary relations rather than systems of segments. It is quite remarkable that traditional graph theory is often incapable of providing satisfactory answers for the most natural questions concerning the drawings of graphs” [10].

Recall that a directed graph is transitive if and implies . One can view a transitive graph as an alternate way of defining a partial order [13]. By dropping the orientation on the edges of a transitive graph we obtain a comparability graph. The complement of a comparability graph is an incomparability graph. In [11] we introduced the concepts of pseudo transitivity and strong pseudo transitivity which are ways of generalizing the concept of transitivity in graphs. The motivation behind writing this paper arose from recent results in computational geometry [2, 7] and intersection graphs theory [6], [8] that can be unified and generalized using the concept of strong pseudo transitivity.

Let be a finite set and be a collection of subsets of . The intersection (overlap) graph of is a graph with the vertex set , where is an edge if (, and , ).

Many intersecting graphs are the intersection or the overlap graphs of combinatorial or geometric structures. For instance incomparability graphs are intersection graphs of monotone curves intersecting two lines parallel to the axis [4], overlap graphs are overlap graphs of intervals on a line, chordal graphs are the intersection graph of subtrees of a tree [9], polygon-circle graphs are the intersection graphs of convex polygons in a circle, Interval filament graphs are the intersection graphs of filaments over intervals [8] and, subtree filament are intersection graphs of filaments over trees [8], or overlap graph of subtrees of a tree [6].

A directed graph is pseudo transitive if there is a partition of so that graphs is transitive, and additionally, if and implies that . A directed graph is strongly pseudo transitive if there is a partition of so that graphs and are transitive, and additionally, if and implies that . Let be pseudo transitive (pseudo transitive) with the underlying partition , then is pseudo transitive of the first type (pseudo transitive of the first type) if and implies that . An undirected graph is co-pseudo transitive (co-pseudo transitive of the first type) if the complement of has an orientation that is pseudo transitive (pseudo transitive of the first type one. is co-strongly pseudo transitive (co-strongly transitive of the first type), if the complement of has an orientation which is strongly pseudo transitive (co-strongly transitive of the first type). Co-pseudo transitive graphs contain the intersection graph of many geometric structures. For instance, the following result was shown in [11], with a slightly altered language.

###### Theorem 1.1

Let be a finite collection of bounded closed subsets of , then the intersection graph of is co-pseudo transitive of the first type.

A half segment is a straight line segment that has one end point on the -axis, another end point in the upper half plane, and makes an acute angel with axis. Motivation behind introducing these segments arose from the work of Pach and Torocsik [14] on geometric graphs. Biro and Trotter [1] studied properties of partial orders arising from half segments. Computing the maximum independent set in the intersection graph of a set of rectangles is a fundamental problem arising in map labeling [15]. Since the general version of this problem is known to be NP-hard, some researchers have focused to solve the special version of the problem, including the instances where all rectangles are intersected by a diagonal line from below. See the work of Lubiw [7], and Correa, Feuilloley,Perez-Lantero , and Soto [2], which provide an and time algorithm, respectively.

In this paper we explore the connections between the graph classes mentioned above and the class of co-strongly pseudo transitive graphs. Specifically we show that the intersection graphs of half line segments and axis parallel rectangles intersecting a diagonal line from bottom are co-strongly pseudo transitive. Moreover, we show that the class of the interval filament graphs is co-strongly transitive of the first type, and hence the class of polygon circle graphs which is contained in the class of interval filament graphs (but contains the classes of chordal, circular arc, circle, and outer planar graphs), and the class of incomparability graphs are strongly transitive of the first type. For the class of chordal graphs which is contained in the class of polygon circle graphs, we provide two direct direct proofs, showing that they are co-strongly transitive of the first type. Additionally, we present some results concerning the Containment of different classes. A contribution of our work is to connect and unify the problems in computational geometry [2, 7], intersection graph theory [8, 6] and combinatorics [1] using the notion of strong pseudo transitivity, thereby, showing they are all amenable to the algorithmic framework in [11] for solving the maximum independent set in time, thereby, avoiding case by case or lengthy arguments for each scenario.

## 2 Structural Results

###### Theorem 2.1

Let be a set of axis parallel rectangles in the plane all of them are intersected by a diagonal line with the property that if two elements of intersect, then they also intersect below . Let be the intersection graph of , then is co-strongly pseudo transitive.

Proof. For any , let and denote the smallest coordinate, and smallest coordinate of four corners of , respectively. Now let with , so that (the case is symmetric). If , then, orient from to , and place it in . Otherwise if , then, still orient from to but place it in . It can be verified that , and that any non edge of has a orientation in . Furthermore, it can be shown that this case, the directed acyclic graph is strongly pseudo-transitive, with the partition . .

The following result was mentioned in [11] without an explicit proof. Next, we specifically state and prove it.

###### Theorem 2.2

Let be a set of half segments in the plane, and let be the intersection graph of , then is co-strongly pseudo transitive.

Proof. For any , let and denote the smallest coordinate, and the largest coordinate of any points in . Now let so that so that . if , then orient from to and place in . Otherwise, if (note the assumption ), then still orient from to , but place the directed edge in . The remaining of the proof copies previous the theorem.

###### Theorem 2.3

Let be an interval filament graph, then is co-strongly pseudo transitive of the first type.

Proof. Consider a representation of , where is a set of intervals on the real line, and for each is a collection of curves on the half plane above that connects the end points of . Then and furthermore , if for some , and and intersect. Now let . If , then orient from to and place in . Otherwise for some . Since and do not intersect and connect the endpoint for interval , the area under one of them (say ) contains the area under the other (say ). In this case orient from to and place it in . Note that , every has an orientation in . Furthermore the directed acyclic graph , is strongly pseudo transitive, and for any and , we have . .

###### Conjecture 2.1

The class of interval filament graphs is properly contained in the class co-strongly pseudo transitive of the first type.

Since chordal graphs play an important role in graph theory, we give two different direct proofs, based on different representations, showing that they are co-strongly pseudo transitive. The first proof uses the characterization that every chordal graph is the intersection graph of subtrees of the a tree, where, the second assumes a perfect elimination ordering is given.

Let be a perfect elimination ordering (PEO) of a chordal graph . A canonical depth first search tree of (with respect to ) is a depth first search spanning tree rooted at constructed applying the following simple rule for visiting vertices: Assume vertex is currently visited, then, select the next vertex to visit to be the smallest indexed vertex among all unvisited vertices adjacent to in .

###### Theorem 2.4

Every choral graph is co-strongly pseudo transitive of the first type.

First Proof. Let be a tree, let be a set of subtrees of . Assign a root to , embed in the plane, and assign root to each , which is the closest vertex of to . Let be the intersection graph of these subtrees, let be the complement of . To prove the claim we will show there is a suitable orientation on . For any so that is not an ancestor of in , and to the left of in the planar embedding of , orient from to and place the resulting oriented edge in . For any so that is an ancestor of in orient from to and place the resulting oriented edge in .

It can be verified that , every has an orientation in , and that directed acyclic graph , is strongly pseudo transitive, thus verifying the claim. Moreover, in this case and implies .

Second Proof. Let be a PEO of . Let be a canonical depth first search spanning tree of rooted at . Let with . If and are on two different branches of then orient from to and place in . Otherwise, and are on the same branches of . Observe in this case that we must have . Since is a canonical depth first tree, and, orient from to and place in .

Claim. Let be three vertices on the same branch of . If , and , then .

It can verified (using the claim and properties of ) that , every has an orientation in , and that directed acyclic graph is strongly pseudo transitive. Additionally, for any and , we have .

We finish this section by the establishing some containment properties.

###### Theorem 2.5

The class of co-pseudo (co-strongly pseudo) transitive graphs of the first type is contained in the class of co-pseudo (co-strongly pseudo) transitive graphs.

The class of co-strongly pseudo transitive graphs of the first type is properly contained in the class co-pseudo transitive graphs of the first type, unless .

The class of tree filament graphs is contained in the class co-pseudo transitive graphs.

The class of tree filament graphs is properly contained in the class co-pseudo transitive graphs, unless .

Proof. Clearly holds. For , first note that by Theorem 1.1 the intersection graph of a set of rectangles is co-pseudo transitive of the first time. Next note that computing the maximum independent set of is NP-hard, but can be done in time for any co-strongly pseudo transitive graph. We omit proof of . follows that graphs of boxicity two are co-pseudo transitive and computing their maximum independent set is known to be NP-hard, but computing maximum independent set in tree filament graphs can be done in polynomial time. .

## 3 Algorithmic Consequences

The following result was shown in [11]

###### Theorem 3.1

Let be strongly pseudo-transitive. The maximum weighted chain can be computed in .

Using our notations the above theorem implies.

###### Theorem 3.2

Let be co-strongly pseudo-transitive. The maximum weighted independent set can be computed in .

The above theorem implies the following general result.

###### Theorem 3.3

Let be one of the following graphs incomparability overlap , chordal, polygon circle interval filament , or, intersection graph half segments, intersection graph axis parallel rectangles intersected by a diagonal line. Then, the maximum weighted independent set can be computed in .

Note that our general frame work extends the work of Lubiw [7] who showed the weighted maximum independent set of rectangles all which have their right most corner on a line can be computed in time, but gives weaker result than a more recent work of Correa, Feuilloley,Perez-Lantero , Soto [2] that had time complexity.

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