a y b
x1
x2 x3
x4 b, x3
a, x1
x4
x3
x1
x2
Figure 1.9: Explanatory figure to Theorem 1.5.25. The dashed lines are non-edges and the waved lines are paths. A grey edge-labelling symbolizes that this vertex misses the path. A grey edge from a vertex to a path symbolizes that this vertex intercepts every such path.
sets. However, we have already made some attempts to this end and it seems unlikely that such an approach will be successful without some new techniques.
While studying the convexity of AT-free graphs, we resolved an open question from [38]
by proving that any given AT-free graph has an AT-free order that coincides with a BFS order. The proof implies a polynomial time algorithm for the computation of such an order that is at least as fast as the computation of the domination intervals of that graph. As a result, we were able to show that there is a close link between the vertex order characterisation of AT-free graphs, and their characterisation through the spine property. As checking whether a vertex order is an AT-free order is, in fact, as difficult as recognising AT-free graphs, it should still be possible to find AT-free orders in linear time. This could be done by giving a linear time implementation of BFSconv, or by constructing another search scheme with similar structural properties.
Linear vertex orderings of other graph classes, such as interval orderings or cocom-parability orderings, have found many applications in optimisation algorithms on these classes. To the best of our knowledge, no such results are known with respect to AT-free orderings. By using AT-free BFS orderings, such results might be easier to attain. Two of the most likely candidates are the independent set problem and the vertex colouring problem. Should it be possible to compute AT-free orders in linear time, it might even be possible to develop robust optimisation algorithms (see [133]) on AT-free graphs, similar to the maximum clique algorithm on comparability graphs. It is still an open question whether every AT-free graph admits a DFS order whose reversal is AT-free [38].
Finally, we analysed the the Carath´eodory number of the convexity on AT-free graphs.
While we were not able to give a bound on this number, our results suggest that this bound is 3. We have made some progress in proving this conjecture. However, there still remains much to be shown. Due to the nature of the Carath´eodory number, this would imply that AT-free graphs are in some way “two-dimensional”, in the same sense that interval graphs are “one-dimensional” as intersection graphs of intervals on a line.
This raises the question whether AT-free graphs can be characterised using some form of intersection model which mirrors this “two-dimensionality”.
We have seen in the previous chapter that for nearly all of the presented graph convexities the class of graphs having a corresponding halfspace ordering is easily characterised. For trees it was shown that these graphs are exactly the claw-free trees, i.e., the paths. For chordal and interval graphs this defines exactly the class of unit interval graphs. In the case of AT-free graphs, however, this question is not as easy to resolve.
In this chapter, we will show that the class of graphs having an AT-free halfspace ordering, which from now on will be called bilateral AT-free, is difficult to characterise and is, in fact,N P-hard to recognise. Note that in this chapter we will use mainlyopen domination intervals, as it makes notation easier here.
Definition 2.0.1. A graph G = (V, E) is bilateral asteroidal triple free if and only if there exists an ordering σ of V such that for any triple a, b, c where a ∈Idom(b, c), we have b≺σ a≺σ c.
We will show how this class relates to other known subclasses of AT-free graphs.
Furthermore, using results from the previous chapter, we present some subclasses of AT-free graphs which we show to be bilateral AT-AT-free. For these classes, we present linear time algorithms to compute a bilateral AT-free ordering.
2.1 Subfamilies of AT-free Graphs
Before we turn to the problem of recognising bilateral AT-free graphs, we will first compare this class to other known subfamilies of AT-free graphs. Figure 2.1a shows a graph which is AT-free and not bilateral AT-free. This shows that the family of bilateral AT-free graphs is a true subset of AT-free graphs. However, this class still contains important subclasses of AT-free graphs.
Lemma 2.1.1 (Mouatadid [116]). Every cocomparability ordering is a bilateral AT-free ordering, i.e., every cocomparability graph is also a bilateral AT-free graph.
Proof. Suppose there exists a graphGwhich has a cocomparability orderingσthat is not a bilateral AT-free ordering. In particular, there exist z, a, b∈V(G) withz∈Idom(a, b) such that z ≺σ a ≺σ b. Let Q be the z−a path avoiding b and let P be the z−b path avoiding a. As P is ab-z path andz≺σ a≺σ b, there exists an edge uv ∈E such that u≺σ a≺σ v andua, va /∈E. This contradicts the fact that σ is a cocomparability ordering.
As, for example, the C5 is bilateral AT-free but not a cocomparability graph, we can state the following corollary.
(a) Graph that is AT-free but not bilateral AT-free.
(b) Graph that is bilateral AT-free but neither path-orderable nor strong asteroid free.
Figure 2.1: Examples of AT-free graphs that differentiate between the subclasses.
Corollary 2.1.2. The class of cocomparability graphs is strictly contained in the class of bilateral AT-free graphs.
In an attempt to characterise AT-free graphs using linear vertex orderings, Corneil et al. [42] introduced the following subfamily of AT-free graphs.
Definition 2.1.3. A graph G= (V, E) is path-orderable if and only if there exists an ordering σ of V, such that for any three vertices a, b, c ∈ V, where a ≺σ b ≺σ c and ac /∈E, any a-c-path contains at least one neighbour of b. Such an ordering is called a path order.
This family can also be shown to be a subfamily of bilateral AT-free graphs.
Lemma 2.1.4 (Mouatadid [116]). Every path-ordering is a bilateral AT-free ordering, i.e., every path-orderable graph is bilateral AT-free
Proof. Suppose there exists a graphGwhich has a path-orderingσ that is not a bilateral AT-free ordering. In particular, there existz, a, b∈V(G) with z∈Idom(a, b) such that z ≺σ a≺σ b. Since σ is a path-ordering, any z-b-path must contain a neighbour of a;
this is a contradiction to the fact thatz∈Idom(a, b).
The graph shown in Figure 2.1b is not a path orderable graph, but it does have a bilateral AT-free ordering, implying the following.
Corollary 2.1.5. The class of path orderable graphs is strictly contained in the class of bilateral AT-free graphs.
Furthermore, we wish to compare bilateral AT-free graphs with the class of strong asteroid free graphs, a class that was defined by Corneil et al. [42] as a polynomially recognisable superclass of path orderable graphs. The definition of these graphs is quite involved and probably best understood in context. Therefore, we will defer it to Sec-tion 2.2.1. Summing up the relaSec-tions between the analysed graph classes, we get:
• cocomparability (path orderable( strong asteroid free(AT-free
• cocomparability (path orderable( bilateral AT-free(AT-free
• strong asteroid free (? bilateral AT-free This leaves us with the following question.
Question 2.1.6. Does the class of bilateral AT-free graphs contain strong asteroid free graphs or are the two classes incomparable?
To answer this question we first present a characterisation of bilateral AT-free graphs which will also lead to a proof of the N P-completeness of their recognition.