Weak Bipolarizable Graphs

We characterize a new class of perfectly orderable graphs and give a polynomial-time recognition algorithm, together with linear-time optimization algorithms for this class of graphs

Recognition of some perfectly orderable graph classes

AbstractThis paper presents new algorithms for recognizing several classes of perfectly orderable graphs. Bipolarizable and P4-simplicial graphs are recognized in O(n3.376) time, improving the previous bounds of O(n4) and O(n5), respectively. Brittle and semi-simplicial graphs are recognized in O(n3) time using a randomized algorithm, and O(n3log2n) time if a deterministic algorithm is required. The best previous time bound for recognizing these classes of graphs is O(m2). Welsh–Powell opposition graphs are recognized in O(n3) time, improving the previous bound of O(n4). HHP-free graphs and maxibrittle graphs are recognized in O(mn) and O(n3.376) time, respectively

Linear Time Optimization Algorithms for P4-Sparse Graphs

Quite often, real-life applications suggest the study of graphs that feature some local density properties. In particular, graphs that are unlikely to have more than a few chordless paths of length three appear in a number of contexts. A graph G is P4-sparse if no set of five vertices in G induces more than one chordless path of length three. P4-sparse graphs generalize both the class of cographs and the class of P4-reducible graphs. It has been shown that P4-sparse graphs can be recognized in time linear in the size of the graph. The main contribution of this paper is to show that once the data structures returned by the recognition algorithm are in place, a number of NP-hard problems on general graphs can be solved in linear time for P4-sparse graphs. Specifically with an n-vertex P4-sparse graph as input the problems of finding a maximum size clique, maximum size stable set, a minimum coloring, a minimum covering by clique, and the size of the minimum fill-in can be solved in O(n) time, independent of the number of edges in the graph

Some Aspects of the Semi-Perfect Elimination

Several efficient algorithms have been proposed to construct a perfect elimination ordering of the vertices of a chordal graph. We study the behaviour of two of these algorithms in relation to a new concept, namely the semi-perfect elimination ordering, which provides a natural generalization of chordal graphs

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

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.

Quasi-Brittle Graphs, a New Class of Perfectly Orderable Graphs

A graph G is quasi-brittle if every induced subgraph H of G contains a vertex which is incident to no edge extending symmetrically to a chordless path with three edges in either Hor its complement H¯. The quasi-brittle graphs turn out to be a natural generalization of the well-known class of brittle graphs. We propose to show that the quasi-brittle graphs are perfectly orderable in the sense of Chvátal: there exists a linear order \u3c on their set of vertices such that no induced path with vertices a, b, c, d and edges ab, bc, cd has a \u3c b and d \u3c c

Algorithmes polynomiaux paramétrés pour des classes de graphes de largeur de clique bornée

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