Practical and Efficient Split Decomposition via Graph-Labelled Trees

Split decomposition of graphs was introduced by Cunningham (under the name join decomposition) as a generalization of the modular decomposition. This paper undertakes an investigation into the algorithmic properties of split decomposition. We do so in the context of graph-labelled trees (GLTs), a new combinatorial object designed to simplify its consideration. GLTs are used to derive an incremental characterization of split decomposition, with a simple combinatorial description, and to explore its properties with respect to Lexicographic Breadth-First Search (LBFS). Applying the incremental characterization to an LBFS ordering results in a split decomposition algorithm that runs in time $O(n+m)\alpha(n+m)$, where $\alpha$ is the inverse Ackermann function, whose value is smaller than 4 for any practical graph. Compared to Dahlhaus' linear-time split decomposition algorithm [Dahlhaus'00], which does not rely on an incremental construction, our algorithm is just as fast in all but the asymptotic sense and full implementation details are given in this paper. Also, our algorithm extends to circle graph recognition, whereas no such extension is known for Dahlhaus' algorithm. The companion paper [Gioan et al.] uses our algorithm to derive the first sub-quadratic circle graph recognition algorithm

Computing commons interval of K permutations, with applications to modular decomposition of graphs

International audienceWe introduce a new way to compute common intervals of K permutations based on a very simple and general notion of generators of common intervals. This formalism leads to simple and efficient algorithms to compute the set of all common intervals of K permutations, that can contain a quadratic number of intervals, as well as a linear space basis of this set of common intervals. Finally, we show how our results on permutations can be used for computing the modular decomposition of graphs in linear time

Linear Time Recognition of P4-Indifferent Graphs

A simple graph is P4-indifferent if it admits a total order b > c > d. P4-indifferent graphs generalize indifferent graphs and are perfectly orderable. Recently, Hoang,Maray and Noy gave a characterization of P4-indifferent graphs interms of forbidden induced subgraphs. We clarify their proof and describe a linear time algorithm to recognize P4-indifferent graphs. Whenthe input is a P4-indifferent graph, then the algorithm computes an order < as above.Key words: P4-indifference, linear time, recognition, modular decomposition.

A survey on algorithmic aspects of modular decomposition

The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important preprocessing step to solve a large number of combinatorial optimization problems. Since the first polynomial time algorithm in the early 70's, the algorithmic of the modular decomposition has known an important development. This paper survey the ideas and techniques that arose from this line of research

Fully polynomial FPT algorithms for some classes of bounded clique-width graphs

Parameterized complexity theory has enabled a refined classification of the difficulty of NP-hard optimization problems on graphs with respect to key structural properties, and so to a better understanding of their true difficulties. More recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as: Strong Exponential Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms, nor even truly subcubic algorithms (Williams and Williams, FOCS 2010 and Abboud, Grandoni, Williams, SODA 2015). A central technique used to tackle the difficulty of the above mentioned problems is fixed-parameter algorithms for polynomial-time problems with polynomial dependency in the fixed parameter (P-FPT). This technique was introduced by Abboud, Williams and Wang in SODA 2016 and continued by Husfeldt (IPEC 2016) and Fomin et al. (SODA 2017), using the treewidth as a parameter. Applying this technique to clique-width, another important graph parameter, remained to be done. In this paper we study several graph theoretic problems for which hardness results exist such as cycle problems (triangle detection, triangle counting, girth, diameter), distance problems (diameter, eccentricities, Gromov hyperbolicity, betweenness centrality) and maximum matching. We provide hardness results and fully polynomial FPT algorithms, using clique-width and some of its upper-bounds as parameters (split-width, modular-width and $P\_4$-sparseness). We believe that our most important result is an ${\cal O}(k^4 \cdot n + m)$-time algorithm for computing a maximum matching where $k$ is either the modular-width or the $P\_4$-sparseness. The latter generalizes many algorithms that have been introduced so far for specific subclasses such as cographs, $P\_4$-lite graphs, $P\_4$-extendible graphs and $P\_4$-tidy graphs. Our algorithms are based on preprocessing methods using modular decomposition, split decomposition and primeval decomposition. Thus they can also be generalized to some graph classes with unbounded clique-width

Algorithmic Aspects of a General Modular Decomposition Theory

A new general decomposition theory inspired from modular graph decomposition is presented. This helps unifying modular decomposition on different structures, including (but not restricted to) graphs. Moreover, even in the case of graphs, the terminology ``module'' not only captures the classical graph modules but also allows to handle 2-connected components, star-cutsets, and other vertex subsets. The main result is that most of the nice algorithmic tools developed for modular decomposition of graphs still apply efficiently on our generalisation of modules. Besides, when an essential axiom is satisfied, almost all the important properties can be retrieved. For this case, an algorithm given by Ehrenfeucht, Gabow, McConnell and Sullivan 1994 is generalised and yields a very efficient solution to the associated decomposition problem

Resolving Prime Modules: The Structure of Pseudo-cographs and Galled-Tree Explainable Graphs

The modular decomposition of a graph $G$ is a natural construction to capture key features of $G$ in terms of a labeled tree $(T,t)$ whose vertices are labeled as "series" ($1$), "parallel" ($0$) or "prime". However, full information of $G$ is provided by its modular decomposition tree $(T,t)$ only, if $G$ is a cograph, i.e., $G$ does not contain prime modules. In this case, $(T,t)$ explains $G$, i.e., $\{x,y\}\in E(G)$ if and only if the lowest common ancestor $\mathrm{lca}_T(x,y)$ of $x$ and $y$ has label "$1$". Pseudo-cographs, or, more general, GaTEx graphs $G$ are graphs that can be explained by labeled galled-trees, i.e., labeled networks $(N,t)$ that are obtained from the modular decomposition tree $(T,t)$ of $G$ by replacing the prime vertices in $T$ by simple labeled cycles. GaTEx graphs can be recognized and labeled galled-trees that explain these graphs can be constructed in linear time. In this contribution, we provide a novel characterization of GaTEx graphs in terms of a set $\mathfrak{F}_{\mathrm{GT}}$ of 25 forbidden induced subgraphs. This characterization, in turn, allows us to show that GaTEx graphs are closely related to many other well-known graph classes such as $P_4$-sparse and $P_4$-reducible graphs, weakly-chordal graphs, perfect graphs with perfect order, comparability and permutation graphs, murky graphs as well as interval graphs, Meyniel graphs or very strongly-perfect and brittle graphs. Moreover, we show that every GaTEx graph as twin-width at most 1.Comment: 18 pages, 3 figure