Michael U. Gerber

Algorithms for vertex-partitioning problems on graphs with fixed clique-width

Many vertex-partitioning problems can be expressed within a general framework introduced by Telle and Proskurowski. They showed that optimization problems in this framework can be solved in polynomial time on classes of graphs with bounded tree-width. In this paper, we consider a very similar framework, in relationship with more general classes of graphs: we propose a polynomial time algorithm on classes of graphs with bounded clique-width for all the optimization problems in our framework. These classes of graphs are more general than the classes of graphs with bounded tree-width in the sense that classes of graphs with bounded tree-width have also bounded clique-width (but not necessarily the inverse).Our framework includes problems such as independent (dominating) set, p-dominating set, induced bounded degree subgraph, induced p-regular subgraph, perfect matching cut, graph k-coloring and graph list-k-coloring with cardinality constraints (fixed k). This paper thus provides a second (distinct) framework within which the optimization problems can be solved in polynomial time on classes of graphs with bounded clique-width, after a first framework (called MS1) due to the work of Courcelle, Makowsky and Rotics (for which they obtained a linear time algorithm).

On the stable set problem in special P 5 -free graphs

The complexity status of the stable set problem in P5-free graphs remains an open question for a long time in spite of a lot of particular results in this direction. The purpose of the present paper is to summarize these results and to propose several new ones. In particular, we prove that the problem of finding a maximum stable set can be solved in polynomial time in the class of (P5,Km,m)-free graphs for any fixed m.

Algorithmic approach to the satisfactory graph partitioning problem

Abstract In a given graph, we want to partition the set of its vertices in two subsets, such that each vertex is satisfied in that it has at least as many neighbours in its own subset as in the other. By introducing weights for the vertices and the edges, we generalize the problem. These more general problems are strongly NP-complete. For the unweighted version, we present some sufficient conditions for the existence of a solution, and propose an exact and a heuristic solution method.

Stable sets in two subclasses of banner-free graphs

The maximum stable set problem is NP-hard, even when restricted to banner-free graphs. In this paper, we use the augmenting graph approach to attack the problem in two subclasses of banner-free graphs. We first provide both classes with the complete characterization of minimal augmenting graphs. Based on the obtained characterization, we prove polynomial solvability of the problem in the class of (banner, P8)-free graphs, improving several existing results.

P 5 -free augmenting graphs and the maximum stable set problem

The complexity status of the maximum stable set problem in the class of P5-free graphs is unknown. In this paper, we first propose a characterization of all connected P5-free augmenting graphs. We then use this characterization to detect families of subclasses of P5-free graphs where the maximum stable set problem has a polynomial time solution. These families extend several previously studied classes.

Robust Algorithms for the Stable Set Problem

Abstract.The stable set problem is to find in a simple graph a maximum subset of pairwise non-adjacent vertices. The problem is known to be NP-hard in general and can be solved in polynomial time on some special classes, like cographs or claw-free graphs. Usually, efficient algorithms assume membership of a given graph in a special class. Robust algorithms apply to any graph G and either solve the problem for G or find in it special forbidden configurations. In the present paper we describe several efficient robust algorithms, extending some known results.

On the Jump Number Problem in Hereditary Classes of Bipartite Graphs

We prove a necessary condition for polynomial solvability of the jump number problem in classes of bipartite graphs characterized by a finite set of forbidden induced bipartite subgraphs. For some classes satisfying this condition, we propose polynomial algorithms to solve the jump number problem.

Augmenting chains in graphs without a skew star

The augmenting chain technique has been applied to solve the maximum stable set problem in the class of line graphs (which coincides with the maximum matching problem) and then has been extended to the class of claw-free graphs. In the present paper, we propose a further generalization of this approach. Specifically, we show how to find an augmenting chain in graphs containing no skew star, i.e. a tree with exactly three vertices of degree 1 of distances 1, 2, 3 from the only vertex of degree 3. As a corollary, we prove that the maximum stable set problem is polynomially solvable in a class that strictly contains claw-free graphs, improving several existing results.

A Transformation Which Preserves the Clique Number

We introduce a graph transformation which preserves the clique number. When applied to graphs containing no odd hole and no cricket (a particular graph on 5 vertices) the transformation also preserves the chromatic number. Using this transformation we derive a polynomial algorithm for the computation of the clique number of all graphs in a class which strictly contains diamond-free graphs. Furthermore, the transformation leads to a proof that the Strong Perfect Graph Conjecture is true for two new classes of graphs and yields a polynomial time algorithm for the computation of the clique number and the chromatic number for both classes. One of these two classes strictly contains claw-free graphs.