Separability and Vertex Ordering of Graphs

Many graph optimization problems, such as finding an optimal coloring, or a largest clique, can be solved by a divide-and-conquer approach. One such well-known technique is decomposition by clique separators where a graph is decomposed into special induced subgraphs along their clique separators. While the most common practice of this method employs minimal clique separators, in this work we study other variations as well. We strive to characterize their structure and in particular the bound on the number of atoms. In fact, we strengthen the known bounds for the general clique cutset decomposition and the minimal clique separator decomposition. Graph ordering is the arrangement of a graph’s vertices according to a certain logic and is a useful tool in optimization problems. Special types of vertices are often recognized in graph classes, for instance it is well-known every chordal graph contains a simplicial vertex. Vertex-ordering, based on such properties, have originated many linear time algorithms. We propose to define a new family named SE-Class such that every graph belonging to this family inherently contains a simplicial extreme, that is a vertex which is either simplicial or has exactly two neighbors which are non-adjacent. Our family lends itself to an ordering based on simplicial extreme vertices (named SEO) which we demonstrate to be advantageous for the coloring and maximum clique problems. In addition, we examine the relation of SE-Class to the family of (Even-Hole, Kite)-free graphs and show a linear time generation of SEO for (Even-Hole, Diamond, Claw)-free graphs. We showcase the applications of those two core tools, namely clique-based decomposition and vertex ordering, on the (Even-Hole, Kite)-free family

The b-Matching Problem in Distance-Hereditary Graphs and Beyond

We make progress on the fine-grained complexity of Maximum-Cardinality Matching on graphs of bounded clique-width. Quasi linear-time algorithms for this problem have been recently proposed for the important subclasses of bounded-treewidth graphs (Fomin et al., SODA\u2717) and graphs of bounded modular-width (Coudert et al., SODA\u2718). We present such algorithm for bounded split-width graphs - a broad generalization of graphs of bounded modular-width, of which an interesting subclass are the distance-hereditary graphs. Specifically, we solve Maximum-Cardinality Matching in O((k log^2{k})*(m+n) * log{n})-time on graphs with split-width at most k. We stress that the existence of such algorithm was not even known for distance-hereditary graphs until our work. Doing so, we improve the state of the art (Dragan, WG\u2797) and we answer an open question of (Coudert et al., SODA\u2718). Our work brings more insights on the relationships between matchings and splits, a.k.a., join operations between two vertex-subsets in different connected components. Furthermore, our analysis can be extended to the more general (unit cost) b-Matching problem. On the way, we introduce new tools for b-Matching and dynamic programming over split decompositions, that can be of independent interest

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

Recognizing Graphs Close to Bipartite Graphs with an Application to Colouring Reconfiguration

We continue research into a well-studied family of problems that ask whether the vertices of a given graph can be partitioned into sets A and B, where A is an independent set and B induces a graph from some specified graph class G. We consider the case where G is the class of k-degenerate graphs. This problem is known to be polynomial-time solvable if k = 0 (recognition of bipartite graphs), but NP-complete if k = 1 (near-bipartite graphs) even for graphs of maximum degree 4. Yang and Yuan [DM, 2006] showed that the k = 1 case is polynomial-time solvable for graphs of maximum degree 3. This also follows from a result of Catlin and Lai [DM, 1995]. We study the general k ≥ 1 case for n-vertex graphs of maximum degree k + 2 We show how to find A and B in O(n) time for k = 1, and in O(n 2 ) time for k ≥ 2. Together, these results provide an algorithmic version of a result of Catlin [JCTB, 1979] and also provide an algorithmic version of a generalization of Brook’s Theorem, proved by Borodin, Kostochka and Toft [DM, 2000] and Matamala [JGT, 2007]. The results also enable us to solve an open problem of Feghali et al. [JGT, 2016]. For a given graph G and positive integer `, the vertex colouring reconfiguration graph of G has as its vertex set the set of `-colourings of G and contains an edge between each pair of colourings that differ on exactly on vertex. We complete the complexity classification of the problem of finding a path in the reconfiguration graph between two given `-colourings of a given graph of maximum degree k