Juraj Stacho

Polarity of chordal graphs

Polar graphs are a common generalization of bipartite, cobipartite, and split graphs. They are defined by the existence of a certain partition of vertices, which is NP-complete to decide for general graphs. It has been recently proved that for cographs, the existence of such a partition can be characterized by finitely many forbidden subgraphs, and hence tested in polynomial time. In this paper we address the question of polarity of chordal graphs, arguing that this is in essence a question of colourability, and hence chordal graphs are a natural restriction. We observe that there is no finite forbidden subgraph characterization of polarity in chordal graphs; nevertheless we present a polynomial time algorithm for polarity of chordal graphs. We focus on a special case of polarity (called monopolarity) which turns out to be the central concept for our algorithms. For the case of monopolar graphs, we illustrate the structure of all minimal obstructions; it turns out that they can all be described by a certain graph grammar, permitting our monopolarity algorithm to be cast as a certifying algorithm.

Reduced clique graphs of chordal graphs

We investigate the properties of chordal graphs that follow from the well-known fact that chordal graphs admit tree representations. In particular, we study the structure of reduced clique graphs which are graphs that canonically capture all tree representations of chordal graphs. We propose a novel decomposition of reduced clique graphs based on two operations: edge contraction and removal of the edges of a split. Based on this decomposition, we characterize asteroidal sets in chordal graphs, and discuss chordal graphs that admit a tree representation with a small number of leaves.

On injective colourings of chordal graphs

We show that one can compute the injective chromatic number of a chordal graph G at least as efficiently as one can compute the chromatic number of (G-B)2, where B are the bridges of G. In particular, it follows that for strongly chordal graphs and so-called power chordal graphs the injective chromatic number can be determined in polynomial time. Moreover, for chordal graphs in general, we show that the decision problem with a fixed number of colours is solvable in polynomial time. On the other hand, we show that computing the injective chromatic number of a chordal graph is NP-hard; and unless NP = ZPP, it is hard to approximate within a factor of n1/3-Ɛ, for any Ɛ > 0. For split graphs, this is best possible, since we show that the injective chromatic number of a split graph is 3√n-approximable. (In the process, we correct a result of Agnarsson et al. on inapproximability of the chromatic number of the square of a split graph.).

Recognizing some subclasses of vertex intersection graphs of 0-bend paths in a grid

We investigate graphs that can be represented as vertex intersections of horizontal and vertical paths in a grid, known as B0 -VPG graphs. Recognizing these graphs is an NP-hard problem. In light of this, we focus on their subclasses. In the paper, we describe polynomial time algorithms for recognizing chordal B0 -VPG graphs, and for recognizing B0 -VPG graphs that have a representation on a grid with 2 rows.

Polynomial-Time Algorithm for the Leafage of Chordal Graphs

Every chordal graph G can be represented as the intersection graph of a collection of subtrees of a host tree, the so-called tree model of G. The leafage l(G) of a connected chordal graph G is the minimum number of leaves of the host tree of a tree model of G. This concept was first defined by I.-J. Lin, T.A. McKee, and D.B. West in [9]. In this contribution, we present the first polynomial time algorithm for computing l(G) for a given chordal graph G. In fact, our algorithm runs in time O(n 3) and it also constructs a tree model of G whose host tree has l(G) leaves.

Vertex Ordering Characterizations of Graphs of Bounded Asteroidal Number

Asteroidal Triple-free AT-free graphs have received considerable attention due to their inclusion of various important graphs families, such as interval and cocomparability graphs. The asteroidal number of a graph is the size of a largest subset of vertices such that the removal of the closed neighborhood of any vertex in the set leaves the remaining vertices of the set in the same connected component. AT-free graphs have asteroidal number at most 2. In this article, we characterize graphs of bounded asteroidal number by means of a vertex elimination ordering, thereby solving a long-standing open question in algorithmic graph theory. Similar characterizations are known for chordal, interval, and cocomparability graphs.

Contact Representations of Planar Graphs: Extending a Partial Representation is Hard

Planar graphs are known to have geometric representations of various types, e.g. as contacts of disks, triangles or - in the bipartite case - vertical and horizontal segments. It is known that such representations can be drawn in linear time, we here wonder whether it is as easy to decide whether a partial representation can be completed to a representation of the whole graph. We show that in each of the cases above, this problem becomes NP-hard. These are the first classes of geometric graphs where extending partial representations is provably harder than recognition, as opposed to e.g. interval graphs, circle graphs, permutation graphs or even standard representations of plane graphs.

Max point-tolerance graphs

A graph G is a max point-tolerance (MPT)graph if each vertex v of G can be mapped to a pointed-interval (Iv, pv) where Iv is an interval ofR and pv ∈ Iv such that uv is an edge of G iff Iu∩ Iv ⊇ {pu, pv}. MPT graphs model relationships among DNA fragments in genome-wide association studies as well as basic transmission problems in telecommunications. We formally introduce this graph class, characterize it, study combinatorial optimization problems on it, and relate it to several well known graph classes. We characterize MPT graphs as a special case of several 2D geometric intersection graphs; namely, triangle, rectangle, L-shape, and line segment intersection graphs. We further characterize MPT as having certain linear orders on their vertex set. Our last characterizatio n is that MPT graphs are precisely obtained by intersecting special pairs of interval graphs. We also show that, on MPT graphs, the maximum weight independent set problem can be solved in polynomial time, the coloring problem is NP-complete, and the clique cover problem has a 2-approximation. Finally, we demonstrate several connections to known graph classes; e.g., MPT graphs strictly contain interval graphs and outerplanar graphs, but are incomparable to permutation, chordal, and planar graphs.

3-Colouring AT-Free Graphs in Polynomial Time

Determining the complexity of the colouring problem on AT-free graphs is one of long-standing open problems in algorithmic graph theory. One of the reasons behind this is that AT-free graphs are not necessarily perfect unlike many popular subclasses of AT-free graphs such as interval graphs or co-comparability graphs. In this paper, we resolve the smallest open case of this problem, and present the first polynomial time algorithm for the 3-colouring problem on AT-free graphs.

Constraint Satisfaction with Counting Quantifiers

We initiate the study of constraint satisfaction problems (CSPs) in the presence of counting quantifiers