10 research outputs found
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
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
Results on perfect graphs
The chromatic number of a graph G is the least number of colours that can be assigned
to the vertices of G such that two adjacent vertices are assigned different colours. The
clique number of a graph G is the size of the largest clique that is an induced subgraph
of G. The notion of perfect graphs was first introduced by Claude Berge in 1960. He
defined a graph G to be perfect if the chromatic number of H is equal to the clique
number of H for every induced subgraph H C G. He also conjectured that perfect
graphs are exactly the class of graphs with no induced odd hole (a chordless odd cycle
of greater than or equal to five vertices) or no induced complement of an odd hole, an
odd anti-hole. This conjecture, that still remains an open problem, is better known as the
Strong Perfect Graph Conjecture (or SPGC). An equivalent statement to SPGC is that
minimal imperfect graphs are odd holes and odd anti-holes.
Fonlupt conjectured that all minimal imperfect graphs with a minimal cutset that
is the union of more than one disjoint clique, must be an odd hole. In this thesis we
prove that any hole-free graph G with a minimal cutset C that is the union of vertexdisjoint
cliques must have a clique in each component o f G — C that sees all of C. We
further prove that minimal imperfect graphs with a minimal cutset that is the union of
two disjoint cliques have a hole.
Since the introduction of perfectly orderable graphs by Chvdtal in 1984, many classes
of perfectly orderable graphs and their recognition algorithms have been identified. Perfectly
ordered graphs are those graphs G such that for each induced ordered subgraph
H of G, the greedy (or, sequential) colouring algorithm produces an optimal colouring
of H. Hohng and Reed previously studied six natural subclasses of perfecdy orderable
graphs that are defined by the orientations of the P4 ’s. Four of the six classes can be
recognized in polynomial time. The recognition problem for the fifth class has been
proven to be NP-complete. In this thesis, we discuss the problem o f recognition for the sixth class, known as one-in-one-out graphs. Also, we consider pyramid-free graphs with
the same orientation as one-in-one-out graphs and prove that this class of graphs cannot
contain a directed 3-cycle of more than one equivalence class
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
-sparseness). We believe that our most important result is an -time algorithm for computing a maximum matching where
is either the modular-width or the -sparseness. The latter generalizes
many algorithms that have been introduced so far for specific subclasses such
as cographs, -lite graphs, -extendible graphs and -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
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 -sparseness). We believe that our most important result is an -time algorithm for computing a maximum matching where is either the modular-width or the -sparseness. The latter generalizes many algorithms that have been introduced so far for specific subclasses such as cographs, -lite graphs, -extendible graphs and -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 bipolarizable and P4-simplicial graphs
Abstract. HoĂ ng and Reed defined the classes of Raspail (also known as Bipolarizable) and P4-simplicial graphs, both of which are perfectly orderable, and proved that they admit polynomial-time recognition algorithms [16]. In this paper, we consider the recognition problem on these classes of graphs and present algorithms that solve it in O(nm) time, where n and m are the numbers of vertices and of edges of the input graph. In particular, we prove properties and show that we can produce bipolarizable and P4-simplicial orderings on the vertices of a graph G, if such orderings exist, working only on P3s that participate in P4s of G. The proposed recognition algorithms are simple, use simple data structures and require O(n + m) space. Moreover, we present a diagram on class inclusions and the currently best recognition time complexities for a number of perfectly orderable classes of graphs and some preliminary results on forbidden subgraphs for the class of P4-simplicial graphs. Keywords: Bipolarizable (Raspail) graph, P4-simplicial graph, perfectly orderable graph, recognition, algorithm, complexity, forbidden subgraph.
On the Recognition of Bipolarizable and P4-simplicial Graphs
The classes of Raspail (also known as Bipolarizable) and P4-simplicial graphs were introduced by HoĂ ng and Reed who showed that both classes are perfectly orderable and admit polynomial-time recognition algorithms [16]. In this paper, we consider the recognition problem on these classes of graphs and present algorithms that solve it in O(nm) time. In particular, we prove properties of the graphs investigated and show that we can produce bipolarizable and P4simplicial orderings on the vertices of the input graph G, if such orderings exist, working only on P3s that participate in a P4 of G. The proposed recognition algorithms are simple, use simple data structures and both require O(n + m) space. Additionally, we show how our recognition algorithms can be augmented to provide certificates, whenever they decide that G is not bipolarizable or P4-simplicial; the augmentation takes O(n + m) time and space. Finally, we include a diagram on class inclusions and the currently best recognition time complexities for a number of perfectly orderable classes of graphs