Tınaz Ekim

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.

Construction of sports schedules with multiple venues

A graph theoretical model is presented for constructing calendars for sports leagues where balancing requirement have to be considered with respect to the different venues where competitions are to be located. An inductive construction is given for leagues having a number of teams 2n which is of the form 2^p in particular.

A tutorial on the use of graph coloring for some problems in robotics

We study the problem where a robot has to pick up items of different sizes which are stored along a corridor. A natural requirement is that the items have to be collected in decreasing order of their sizes. We deal with various systems according to the location of the Entry/Exit station where the robot unloads the collected items after each trip along the corridor. The links of these systems with generalized coloring problems and other applications such that train shunting and pallet loading problems are discussed and related results are obtained. We conclude with several open questions on the topic.

Partitioning cographs into cliques and stable sets

We consider the problem of partitioning the node set of a graph into p cliques and k stable sets, namely the (p,k)-coloring problem. Results have been obtained for general graphs [Feder et al., SIAM J. Discrete Math. 16 (3) (2003) 449-478], chordal graphs [Hell et al., Discrete Appl. Math. 141 (2004) 185-194] and cacti for the case where k=p in [Ekim and de Werra, On split-coloring problems, submitted for publication] where some upper and lower bounds on the optimal value minimizing k are also presented. We focus on cographs and devise some efficient algorithms for solving (p,k)-coloring problems and cocoloring problems in O(n^2+nm) time and O(n^3^/^2) time, respectively. We also give an algorithm for finding the maximum induced (p,k)-colorable subgraph. In addition to this, we present characterizations of (2,1)- and (2,2)-colorable cographs by forbidden configurations.

Polar cographs

Polar graphs are a natural extension of some classes of graphs like bipartite graphs, split graphs and complements of bipartite graphs. A graph is (s,k)-polar if there exists a partition A,B of its vertex set such that A induces a complete s-partite graph (i.e., a collection of at most s disjoint stable sets with complete links between all sets) and B a disjoint union of at most k cliques (i.e., the complement of a complete k-partite graph). Recognizing a polar graph is known to be NP-complete. These graphs have not been extensively studied and no good characterization is known. Here we consider the class of polar graphs which are also cographs (graphs without induced path on four vertices). We provide a characterization in terms of forbidden subgraphs. Besides, we give an algorithm in time O(n) for finding a largest induced polar subgraph in cographs; this also serves as a polar cograph recognition algorithm. We examine also the monopolar cographs which are the (s,k)-polar cographs where min(s,k)=<1. A characterization of these graphs by forbidden subgraphs is given. Some open questions related to polarity are discussed.

Minimum maximal matching is NP-hard in regular bipartite graphs

Yannakakis and Gavril showed in [10] that the problem of finding a maximal matching of minimum size (MMM for short), also called Minimum Edge Dominating Set, is NP-hard in bipartite graphs of maximum degree 3 or planar graphs of maximum degree 3. Horton and Kilakos extended this result to planar bipartite graphs and planar cubic graphs [6]. Here, we extend the result of Yannakakis and Gavril in [10] by showing that MMM is NP-hard in the class of k-regular bipartite graphs for all k ≥ 3 fixed.

Polar Permutation Graphs

Polar graphs generalise bipartite, cobipartite, split graphs, and they constitute a special type of matrix partitions. A graph is polar if its vertex set can be partitioned into two, such that one part induces a complete multipartite graph and the other part induces a disjoint union of complete graphs. Deciding whether a given arbitrary graph is polar, is an NP-complete problem. Here we show that for permutation graphs this problem can be solved in polynomial time. The result is surprising, as related problems like achromatic number and cochromatic number are NP-complete on permutation graphs. We give a polynomial-time algorithm for recognising graphs that are both permutation and polar. Prior to our result, polarity has been resolved only for chordal graphs and cographs.

On split-coloring problems

We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this area.

On some applications of the selective graph coloring problem

In this paper we present the Selective Graph Coloring Problem, a generalization of the standard graph coloring problem as well as several of its possible applications. Given a graph with a partition of its vertex set into several clusters, we want to select one vertex per cluster such that the chromatic number of the subgraph induced by the selected vertices is minimum. This problem appeared in the literature under different names for specific models and its complexity has recently been studied for different classes of graphs. Here, we describe different models – some already discussed in previous papers and some new ones – in very different contexts under a unified framework based on this graph problem. We point out similarities between these models, offering a new approach to solve them, and show some generic situations where the selective graph coloring problem may be used. We focus on specific graph classes motivated by each model, and we briefly discuss the complexity of the selective graph coloring problem in each one of these graph classes and point out interesting future research directions.

Recognizing line-polar bipartite graphs in time O(n)

A graph is polar if the vertex set can be partitioned into A and B in such a way that A induces a complete multipartite graph and B induces a disjoint union of cliques (i.e., the complement of a complete multipartite graph). Polar graphs naturally generalize several classes of graphs such as bipartite graphs, cobipartite graphs and split graphs. Recognizing polar graphs is an NP-complete problem in general, and thus it is of interest to restrict the problem to special classes of graphs. Cographs and chordal graphs are among those whose polarity can be recognized in polynomial time. The line-graphs of bipartite graphs are another class of graphs whose polarity has been characterized recently in terms of forbidden subgraphs, but no polynomial time algorithm is given. In this paper, we present an O(n) algorithm which decides whether the line-graph of an input bipartite graph is polar and constructs a polar partition when one exists.