Independent Set Reconfiguration in Cographs

We study the following independent set reconfiguration problem, called TAR-Reachability: given two independent sets $I$ and $J$ of a graph $G$, both of size at least $k$, is it possible to transform $I$ into $J$ by adding and removing vertices one-by-one, while maintaining an independent set of size at least $k$ throughout? This problem is known to be PSPACE-hard in general. For the case that $G$ is a cograph (i.e. $P_4$-free graph) on $n$ vertices, we show that it can be solved in time $O(n^2)$, and that the length of a shortest reconfiguration sequence from $I$ to $J$ is bounded by $4n-2k$, if such a sequence exists. More generally, we show that if $X$ is a graph class for which (i) TAR-Reachability can be solved efficiently, (ii) maximum independent sets can be computed efficiently, and which satisfies a certain additional property, then the problem can be solved efficiently for any graph that can be obtained from a collection of graphs in $X$ using disjoint union and complete join operations. Chordal graphs are given as an example of such a class $X$

Characterizing graph classes by intersections of neighborhoods

The interplay between maxcliques (maximal cliques) and intersections of closed neighborhoods leads to new types of characterizations of several standard graph classes. For instance, being hereditary clique-Helly is equivalent to every nontrivial maxclique $Q$ containing the intersection of closed neighborhoods of two vertices of $Q$, and also to, in all induced subgraphs, every nontrivial maxclique containing a simplicial edge (an edge in a unique maxclique). Similarly, being trivially perfect is equivalent to every maxclique $Q$ containing the closed neighborhood of a vertex of $Q$, and also to, in all induced subgraphs, every maxclique containing a simplicial vertex. Maxcliques can be generalized to maximal cographs, yielding a new characterization of ptolemaic graphs

Bounded Search Tree Algorithms for Parameterized Cograph Deletion: Efficient Branching Rules by Exploiting Structures of Special Graph Classes

Many fixed-parameter tractable algorithms using a bounded search tree have been repeatedly improved, often by describing a larger number of branching rules involving an increasingly complex case analysis. We introduce a novel and general search strategy that branches on the forbidden subgraphs of a graph class relaxation. By using the class of $P_4$-sparse graphs as the relaxed graph class, we obtain efficient bounded search tree algorithms for several parameterized deletion problems. We give the first non-trivial bounded search tree algorithms for the cograph edge-deletion problem and the trivially perfect edge-deletion problems. For the cograph vertex deletion problem, a refined analysis of the runtime of our simple bounded search algorithm gives a faster exponential factor than those algorithms designed with the help of complicated case distinctions and non-trivial running time analysis [21] and computer-aided branching rules [11].Comment: 23 pages. Accepted in Discrete Mathematics, Algorithms and Applications (DMAA

Algorithmic Aspects of Switch Cographs

This paper introduces the notion of involution module, the first generalization of the modular decomposition of 2-structure which has a unique linear-sized decomposition tree. We derive an O(n^2) decomposition algorithm and we take advantage of the involution modular decomposition tree to state several algorithmic results. Cographs are the graphs that are totally decomposable w.r.t modular decomposition. In a similar way, we introduce the class of switch cographs, the class of graphs that are totally decomposable w.r.t involution modular decomposition. This class generalizes the class of cographs and is exactly the class of (Bull, Gem, Co-Gem, C_5)-free graphs. We use our new decomposition tool to design three practical algorithms for the maximum cut, vertex cover and vertex separator problems. The complexity of these problems was still unknown for this class of graphs. This paper also improves the complexity of the maximum clique, the maximum independant set, the chromatic number and the maximum clique cover problems by giving efficient algorithms, thanks to the decomposition tree. Eventually, we show that this class of graphs has Clique-Width at most 4 and that a Clique-Width expression can be computed in linear time