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... 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...
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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... 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.
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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... 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.
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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... 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...
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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... 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.
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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... 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...
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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... 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
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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... 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.
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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... 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 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... 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 interval graphs that have bubble models of specific properties. We characterise the maximal such proper interval graphs of bounded clique-width... 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.
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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... 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.
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We consider three graph partitioning problems, both from the vertices and the edges point of view. These problems are dominating set, list-q-coloring with costs (fixed number of colors q) and coloring with non-fixed number of colors. They... more
We consider three graph partitioning problems, both from the vertices and the edges point of view. These problems are dominating set, list-q-coloring with costs (fixed number of colors q) and coloring with non-fixed number of colors. They are all known to be NP-hard in general. We show that all these problems (except edge-coloring) can be solved in polynomial time on
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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... 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 quantification on vertex sets, that includes NP-hard problems) can be solved efficiently for graphs of small clique-width. It is widely believed that determining the clique-width of a graph is NP-hard; in
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... t t l f l G Pruning tree for G Fig. 1. A connected distance hereditary graph G and a pruning tree for G. (i) Set T1 as the tree consisting of a single root vertex v1, and set i := 1. (ii) Set i := i + 1. If i>n then set T := Tn and... more
... t t l f l G Pruning tree for G Fig. 1. A connected distance hereditary graph G and a pruning tree for G. (i) Set T1 as the tree consisting of a single root vertex v1, and set i := 1. (ii) Set i := i + 1. If i>n then set T := Tn and stop. (iii) Let si = 〈(vi,vj), leaf〉 (resp. si = 〈(vi,vj), f alse〉, or si = ...
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We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is denable in monadic second-order logic (MSOL). We show that for bounded tree-width these problems are solvable in... more
We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is denable 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
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Treewidth is generally regarded as one of the most useful parameterizations of a graph’s construction. Clique-width is a similar parameterizations that shares one of the powerful properties of treewidth, namely: if a graph is of bounded... more
Treewidth is generally regarded as one of the most useful parameterizations of a graph’s construction. Clique-width is a similar parameterizations that shares one of the powerful properties of treewidth, namely: if a graph is of bounded treewidth (or clique-width), then there is a polynomial time algorithm for any graph problem expressible in Monadic Second Order Logic, using quantifiers on vertices
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We consider three graph partitioning problems, both from the vertices and the edges point of view. These problems are dominating set, list-q-coloring with costs (fixed number of colors q) and coloring with non-fixed number of colors. They... more
We consider three graph partitioning problems, both from the vertices and the edges point of view. These problems are dominating set, list-q-coloring with costs (fixed number of colors q) and coloring with non-fixed number of colors. They are all known to be NP-hard in general. We show that all these problems (except edge-coloring) can be solved in polynomial time on
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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... more
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.
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Babel and Olariu (1995) introduced the class of (q, t) graphs in which every set of q vertices has at most t distinct induced P4s. Graphs of clique-width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs... 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 P4s. 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 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 defining 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.
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In this paper, we present the first polynomial time algorithm for recognizing and factoring read-once functions. The algorithm is based on algorithms for cograph recognition and a new efficient method for checking normality. Its... more
In this paper, we present the first polynomial time algorithm for recognizing and factoring read-once functions. The algorithm is based on algorithms for cograph recognition and a new efficient method for checking normality. Its correctness is based on a classical characterization theorem of Gurvich which states that a positive Boolean function is read-once if and only if it is normal
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ABSTRACT A graph is called equistable when there is a nonnegativeweight function on its vertices such that a set S of verticeshas total weight 1 if and only if S is maximal stable. We show thata necessary condition for a graph to be... more
ABSTRACT A graph is called equistable when there is a nonnegativeweight function on its vertices such that a set S of verticeshas total weight 1 if and only if S is maximal stable. We show thata necessary condition for a graph to be equistable is su#cient whenthe graph in question is distance-hereditary. This is used to designa polynomial-time recognition algorithm for equistable distancehereditarygraphs.
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A graph is called equistable when there is a non-negative 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 characterize those series-parallel graphs that are... more
A graph is called equistable when there is a non-negative 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 characterize those series-parallel graphs that are equistable, generalizing results of Mahadev, Peled and Sun about equistable outer-planar graphs.