Clique-width of path powers

Discrete Applied Mathematics, 2012

We study the linear clique-width of graphs that are obtained from paths by disjoint union and adding true twins. We show that these graphs have linear clique-width at most 4, and we give a complete characterisation of their linear clique-width by forbidden induced subgraphs. As a consequence, we obtain a linear-time algorithm for computing the linear clique-width of the considered graphs. Our results extend the previously known set of forbidden induced subgraphs for graphs of linear clique-width at most 3.

2010

Clique-width is an important graph parameter whose computation is NP-hard. In fact we do not know of any other algorithm than brute force for the exact computation of clique-width on any non-trivial graph class. Results so far indicate that proper interval graphs constitute the first interesting graph class on which we might have hope to compute clique-width, or at least its linear variant linear clique-width, in polynomial time. In TAMC 2009, a polynomialtime algorithm for computing linear clique-width on a subclass of proper interval graphs was given. In this paper, we present a polynomial-time algorithm for a larger subclass of proper interval graphs that approximates the clique-width within an additive factor 3. Previously known upper bounds on clique-width result in arbitrarily large difference from the actual clique-width when applied on this class. Our results contribute toward the goal of eventually obtaining a polynomial-time exact algorithm for clique-width on proper interval graphs.

Discrete Applied Mathematics, 2015

Clique-width of graphs is defined algebraically through operations on graphs with vertex labels. We characterise the clique-width in a combinatorial way by means of partitions of the vertex set, using trees of nested partitions where partitions are ordered bottom-up by refinement. We show that the correspondences in both directions, between combinatorial partition trees and algebraic terms, preserve the tree structures and that they are computable in polynomial time. We apply our characterisation to linear clique-width. And we relate our characterisation to a clique-width characterisation by Heule and Szeider that is used to reduce the clique-width decision problem to a satisfiability problem.

Lecture Notes in Computer Science, 2014

We investigate a new width parameter, the fusion-width of a graph. It is a natural generalization of the tree-width, yet strong enough that not only graphs of bounded tree-width, but also graphs of bounded clique-width, trivially have bounded fusion-width. In particular, there is no exponential growth between tree-width and fusion-width, as is the case between tree-width and clique-width. The new parameter gives a good intuition about the relationship between tree-width and cliquewidth.

1999

Babel and Olariu (1995) introduced the class of (q;t) graphs in which every set of q vertices has at most t distinct induced P 4 s. Graphs of clique-width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be de ned by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique{width of the (q;t) graphs, for almost all possible combinations of q and t. On one hand we show that every (q;q ? 3) graph for q 7, has clique{width q and a q{expression de ning it can be obtained in linear time. On the other hand we show that the class of (q;q ?3) graphs for 4 q 6 and the class of (q; q ?1) graphs for q 4 are not of bounded clique{width.

Electronic Notes in Theoretical Computer Science, 2016

Similar to the tree-width (twd), the clique-width (cwd) is an invariant of graphs. A well known relationship between tree-width and clique-width is that cwd(G) ≤ 3 • 2 twd(G)−1. It is also known that tree-width of Cactus graphs is 2, therefore the clique-width for those graphs is smaller or equal than 6. In this paper, it is shown that the clique-width of Cactus graphs is smaller or equal to 4 and we present a polynomial time algorithm which computes exactly a 4-expression.

International Journal of Foundations of Computer Science, 2000

Graphs of clique–width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique–width of perfect graph classes. On one hand, we show that every distance–hereditary graph, has clique–width at most 3, and a 3–expression defining it can be obtained in linear time. On the other hand, we show that the classes of unit interval and permutation graphs are not of bounded clique–width. More precisely, we show that for every [Formula: see text] there is a unit interval graph In and a permutation graph Hn having n2 vertices, each of whose clique–width is at least n. These results allow us to see the border within the hierarchy of perfect graphs between classes whose clique–width is bounded and classes whose clique–width is unbounded. Finally we show that every n×n square grid, [Formula: see text], n ≥ 3, has clique–width exactly n+1.

Discrete Applied Mathematics, 2012

Clique-width is a relatively new parameterization of graphs, philosophically similar to treewidth. Clique-width is more encompassing in the sense that a graph of bounded treewidth is also of bounded clique-width (but not the converse). For graphs of bounded clique-width, given the clique-width decomposition, every optimization, enumeration or evaluation problem that can be defined by a monadic second-order logic formula using quantifiers on vertices, but not on edges, can be solved in polynomial time. This is reminiscent of the situation for graphs of bounded treewidth, where the same statement holds even if quantifiers are also allowed on edges. Thus, graphs of bounded clique-width are a larger class than graphs of bounded treewidth, on which we can resolve fewer, but still many, optimization problems efficiently. One of the major open questions regarding clique-width is whether graphs of cliquewidth at most k, for fixed k, can be recognized in polynomial time. In this paper, we present the first polynomial-time algorithm (O(n 2 m)) to recognize graphs of clique-width at most 3.

Given a graph G, the graph G l has the same vertex set and two vertices are adjacent in G l if and only if they are at distance at most l in G. The l-coloring problem consists in finding an optimal vertex coloring of the graph G l , where G is the input graph. We show that, for any fixed value of l, the l-coloring problem is polynomial when restricted to graphs of bounded NLC-width (or clique-width), if an expression of the graph is also part of the input. We also prove that the NLC-width of G l is at most 2(l + 1) nlcw(G) .

2005

Clique-width is a graph parameter, defined by a composition mechanism for vertexlabeled graphs, which measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in Monadic Second Order Logic, that includes NPhard problems) can be solved efficiently for graphs of certified small clique-width. It is widely believed that determining the clique-width of a graph is NP-hard; in spite of considerable efforts, no NP-hardness proof has been found so far. In this paper we show a non-approximability result for restricted form of cliquewidth, termed "r-sequential clique-width", considering only such clique-width constructions where one of any two graphs put together by disjoint union must have r or fewer vertices. In particular, we show that for every positive integer r, the r-sequential cliquewidth cannot be absolutely approximated in polynomial time unless P = NP, and that given G and k the question of whether the r-sequential clique-width of G is at most k is NP-complete. We show further that this non-approximability result holds even for graphs of a very particular structure: for graphs obtained from cobipartite graphs by replacing edges with induced paths. In part II of this series of papers we use this strengthened result to show that, unless P = NP, there is no polynomial-time absolute approximation algorithm for (unrestricted) clique-width, and that, given a graph G and an integer k, deciding whether the the clique-width G is at most k is NP-complete. This solves a problem that has been open since the introduction of clique-width in the early 1990s.