We discuss the parametrized complexity of counting and evaluation problems on graphs where the ra... more We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is definable in Monadic Second Order Logic. We show that for bounded tree-width these problems are solvable in polynomial time. The same holds for bounded clique width in the cases, where the decomposition, which establishes the bound on the clique--width, can be computed in polynomial time and for problems expressible by Monadic Second Order formulas without edge set quantification. Such quantifications are allowed in the case of graphs with bounded tree-width. As applications we discuss in detail how this affects the parametrized complexity of the permanent and the hamiltonian of a matrix, and more generally, various generating functions of MSOL definable graph properties. Finally, our results are also applicable to SAT and ]SAT . Key words: Fixed parameter complexity, combinatorial enumeration 1 supported by the European project GETGRATS. 2 partially supported by a G...
Factoring and Recognition of Read-Once Functions using Cographs and Normality and the Readability of Functions Associated with Partial k-trees
An approach for factoring general boolean functions was described in [15] which is based on graph... more An approach for factoring general boolean functions was described in [15] which is based on graph partitioning algorithms. In this paper, we present a very fast algorithm for recognizing and factoring readonce functions which is needed as a dedicated factoring subroutine to handle the lower levels of that factoring process. The algorithm is based on algorithms for cograph recognition and on checking normality. For non-read-once functions, we investigate their factoring based on their corresponding graph classes. In particular, we show that if a function F is normal and its corresponding graph is a partial k-tree, then F is a read 2 k function and a read 2 k formula for F can be obtained in polynomial time.
A graph is called equistable when there is a nonnegative weight function on its vertices such tha... more A graph is called equistable when there is a nonnegative weight function on its vertices such that a set S of vertices has total weight 1 if and only if S is maximal stable. We show that a necessary condition for a graph to be equistable is su#cient when the graph in question is distance-hereditary. This is used to design a polynomial-time recognition algorithm for equistable distancehereditary graphs.
Computing the clique-width of large path powers in linear time via a new characterisation of clique-width
Clique-width is one of the most important graph parameters, as many NP-hard graph problems are so... more 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 gra...
We discuss the parametrized complexity of counting and evaluation problems on graphs where the ra... more We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is de nable in monadic second-order logic (MSOL). We show that for bounded tree-width these problems are solvable in polynomial time. The same holds for bounded clique width in the cases, where the decomposition, which establishes the bound on the clique-width, can be computed in polynomial time and for problems expressible by monadic second-order formulas without edge set quanti cation. Such quanti cations are allowed in the case of graphs with bounded tree-width. As applications we discuss in detail how this a ects the parametrized complexity of the permanent and the hamiltonian of a matrix, and more generally, various generating functions of MSOL de nable graph properties. Finally, our results are also applicable to SAT and ]SAT . ? 2001 Elsevier Science B.V. All rights reserved.
Clique-width is a graph parameter, dened by a composition mechanism for vertexlabeled graphs, whi... more Clique-width is a graph parameter, dened 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 ecien tly for graphs of certied small clique-width. It is widely believed that determining the clique-width of a graph is NP-hard; in spite of considerable eorts, 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. We show further that this non-approximability result holds even for graphs of a very part...
10 edges are bounded by 2n, since the degree of each vertex x not chosen at stpe (ii) above is at... more 10 edges are bounded by 2n, since the degree of each vertex x not chosen at stpe (ii) above is at most 2. Thus the nummber of edges in E 0 is bounded by 2n+m?1 which is bounded by 3n. However verifying that G 0 is an optimal 2{spanner is diicult and will not be presented here. 2
Clique-width is a graph parameter that measures in a certain sense the complexity of a graph. Har... more 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 quantication on vertex sets, that includes NP-hard problems) can be solved ecien tly for graphs of certied small clique-width. It is widely believed that determining the clique-width of a graph is NP-hard; in spite of considerable eorts, no NP-hardness proof has been found so far. We give the rst 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.
In the study of full bubble model graphs of bounded clique-width and bounded linear clique-width,... more In the study of full bubble model graphs of bounded clique-width and bounded linear clique-width, we determined complete sets of forbidden induced subgraphs, that are minimal in the class of full bubble model graphs. In this note, we show that (almost all of) these graphs are minimal in the class of all graphs. As a corollary, we can give sets of minimal forbidden induced subgraphs for graphs of bounded clique-width and for graphs of bounded linear clique-width for arbitrary bounds.
A triangle-free graph G is called read-k when there exists a monotone Boolean formulawhose variab... more A triangle-free graph G is called read-k when there exists a monotone Boolean formulawhose variables are the ver- tices of G and whose minterms are precisely the edges of G, such that no variable occurs more than k times in �. The smallest such k is called the readability of G. We exhibit a very simple class of bi- partite chain graphs on 2n vertices with readability �q log n log log n �
A bubble model is a 2-dimensional representation of proper interval graphs. We consider proper in... more A bubble model is a 2-dimensional representation of proper interval graphs. We consider proper interval graphs that have bubble models of specific properties. We characterise the maximal such proper interval graphs of bounded clique-width and of bounded linear cliquewidth and the minimal such proper interval graphs whose clique-width and linear cliquewidth exceed the bounds. As a consequence, we can efficiently compute the clique-width and linear clique-width of the considered graphs.
We describe the clique-width of path powers by an exact formula, depending only on the number of ... more We describe the clique-width of path powers by an exact formula, depending only on the number of vertices and the clique number. As a consequence, the clique-width of path powers can be computed in linear time. Path powers are a graph class of unbounded cliquewidth. Prior to our result, square grids constituted the only known graph class of unbounded clique-width with a similar result. We also show that clique-width and linear clique-width coincide on path powers.
In the study of full bubble model graphs of bounded clique-width and bounded linear clique-width,... more In the study of full bubble model graphs of bounded clique-width and bounded linear clique-width, we determined complete sets of forbidden induced subgraphs, that are minimal in the class of full bubble model graphs. In this note, we show that (almost all of) these graphs are minimal in the class of all graphs. As a corollary, we can give sets of minimal forbidden induced subgraphs for graphs of bounded clique-width and for graphs of bounded linear clique-width for arbitrary bounds.
Babel and Olariu (1995) introduced the class of (q;t) graphs in which every set of q vertices has... more 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.
A triangle-free graph G is called read-k when there exists a monotone Boolean formula φ whose var... more A triangle-free graph G is called read-k when there exists a monotone Boolean formula φ whose variables are the vertices of G and whose minterms are precisely the edges of G, such that no variable occurs more than k times in φ. The smallest such k is called the readability of G. We exhibit a very simple class of bipartite chain graphs on 2n vertices with readability Ω log n log log n .
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