Polynomial-time recognition of clique-width ≤3 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.

2005

Clique-width is a graph parameter that measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in Monadic Second Order Logic with second-order quantification on vertex sets, that includes NP-hard 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. We give the first hardness proof. We show that the clique-width of a given graph cannot be absolutely approximated in polynomial time unless P = NP. We also show that, given a graph G and an integer k, deciding whether the clique-width of 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.

Theoretical Computer Science, 2009

Consider finite, simple and undirected graphs. V and E denote the vertex set and the edge set of the graph G, respectively. A complete set of G is a subset of V inducing a complete subgraph. A clique is a maximal complete set. The clique family of G is denoted by C(G). The clique graph of G is the intersection graph of C(G). The clique operator, K, assigns to each graph G its clique graph which is denoted by K(G). On the other hand, say that G is a clique graph if G belongs to the image of the clique operator, i.e. if there exists a graph H such that G = K (H). Clique operator and its image were widely studied. First articles focused on recognizing clique graphs [20,36], In [4,13], graphs for which the clique graph changes whenever a vertex is removed are considered. Graphs fixed under the operator K or fixed under the iterated clique operator, I<n, for some positive integer n; and the behavior under these operators of parameters such as the number of vertices or diameter were studied in [5,8,9,12,26,30] and more recently in [7,14,21-23, 29], For several classes of graphs, the image of the class under the clique operator was characterized [10,18,19,24,34,37]; and, in some cases, also the inverse image of the class [16,28,35], Results of the previous bibliography can be found in the survey [39], Clique graphs have been much studied as intersection graphs and are included in several books [11,25,33], In this paper we are concerned with the time complexity of the problem of recognizing clique graphs, this is the time complexity of the following decision problem.

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.

Computer Science – Theory and Applications, 2011

Clique-width is one of the most important graph parameters, as many NP-hard graph problems are solvable in linear time on graphs of bounded clique-width. Unfortunately the computation of clique-width is among the hardest problems. In fact we do not know of any other algorithm than brute force for the exact computation of clique-width on any nontrivial large graph class. Another difficulty about clique-width is the lack of alternative characterisations of it that might help in coping with its hardness. In this paper we present two results. The first is a new characterisation of clique-width based on rooted binary trees, completely without the use of labelled graphs. Our second result is the exact computation of the clique-width of large path powers in polynomial time, which has been an open problem since the 1990's. The presented new characterisation is used to achieve this latter result. With our result, large k-path powers constitute the first non-trivial infinite class of graphs of unbounded clique-width whose clique-width can be computed exactly in polynomial time.

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.

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.

ACM Transactions on Algorithms

Recently, hardness results for problems in P were achieved using reasonable complexity-theoretic assumptions such as the Strong Exponential Time Hypothesis. According to these assumptions, many graph-theoretic problems do not admit truly subquadratic algorithms. A central technique used to tackle the difficulty of the above-mentioned problems is fixed-parameter algorithms with polynomial dependency in the fixed parameter (P-FPT). Applying this technique to clique-width , an important graph parameter, remained to be done. In this article, we study several graph-theoretic problems for which hardness results exist such as cycle problems , distance problems , and maximum matching . We give hardness results and P-FPT algorithms, using clique-width and some of its upper bounds as parameters. We believe that our most important result is an algorithm in O ( k 4 ⋅ n + m )-time for computing a maximum matching, where k is either the modular-width of the graph or the P 4 -sparseness. The latte...

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.

Lecture Notes in Computer Science, 2010

Boolean-width is a recently introduced graph invariant. Similar to tree-width, it measures the structural complexity of graphs. Given any graph G and a decomposition of G of booleanwidth k , we give algorithms solving a large class of vertex subset and vertex partitioning problems in time O * (2 O(k 2)). We relate the boolean-width of a graph to its branch-width and to the booleanwidth of its incidence graph. For this we use a constructive proof method that also allows much simpler proofs of similar results on rank-width by Oum (JGT 2008). For a random graph on n vertices we show that almost surely its boolean-width is Θ(log 2 n)-setting boolean-width apart from other graph invariants-and it is easy to find a decomposition witnessing this. Combining our results gives algorithms that on input a random graph on n vertices will solve a large class of vertex subset and vertex partitioning problems in quasi-polynomial time O * (2 O(log 4 n)) .