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18
Booleanwidth of graphs
, 2009
"... We introduce the graph parameter booleanwidth, related to the number of different unions of neighborhoods across a cut of a graph. For many graph problems this number is the runtime bottleneck when using a divideandconquer approach. Booleanwidth is similar to rankwidth, which is related to ..."
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Cited by 13 (8 self)
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We introduce the graph parameter booleanwidth, related to the number of different unions of neighborhoods across a cut of a graph. For many graph problems this number is the runtime bottleneck when using a divideandconquer approach. Booleanwidth is similar to rankwidth, which is related to the number of GF [2]sums (1+1=0) of neighborhoods instead of the Booleansums (1+1=1) used for booleanwidth. For an nvertex graph G given with a decomposition tree of booleanwidth k we show how to solve Minimum Dominating Set, Maximum Independent Set and Minimum or Maximum Independent Dominating Set in time O(n(n+ 23kk)). We show that for any graph the square root of its booleanwidth is never more than its rankwidth. We also exhibit a class of graphs, the Hsugrids, having the property that a Hsugrid on Θ(n2) vertices has booleanwidth Θ(log n) and treewidth, branchwidth, cliquewidth and rankwidth Θ(n). Moreover, any optimal rankdecomposition of such a graph will have booleanwidth Θ(n) , i.e. exponential in the optimal booleanwidth.
Satisfiability of acyclic and almost acyclic CNF formulas
 IN: FSTTCS
, 2010
"... We study the propositional satisfiability problem (SAT) on classes of CNF formulas (formulas in Conjunctive Normal Form) that obey certain structural restrictions in terms of their hypergraph structure, by associating to a CNF formula the hypergraph obtained by ignoring negations and considering cla ..."
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Cited by 10 (4 self)
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We study the propositional satisfiability problem (SAT) on classes of CNF formulas (formulas in Conjunctive Normal Form) that obey certain structural restrictions in terms of their hypergraph structure, by associating to a CNF formula the hypergraph obtained by ignoring negations and considering clauses as hyperedges on variables. We show that satisfiability of CNF formulas with socalled “βacyclic hypergraphs ” can be decided in polynomial time. We also study the parameterized complexity of SAT for “almost” βacyclic instances, using as parameter the formula’s distance from being βacyclic. As distance we use the size of smallest strong backdoor sets and the βhypertree width. As a byproduct we obtain the W[1]hardness of SAT parameterized by the (undirected) cliquewidth of the incidence graph, which disproves a conjecture by Fischer, Makowsky, and Ravve (Discr. Appl. Math. 156, 2008).
On the Band, Tree and CliqueWidth of Graphs With Bounded Vertex Degree
, 2002
"... The band, tree and cliquewidth are of primary importance in algorithmic graph theory due to the fact that many problems being NPhard for general graphs can be solved in polynomial time when restricted to graphs where one of these parameters is bounded. It is known that for any fixed Delta >= ..."
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Cited by 8 (2 self)
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The band, tree and cliquewidth are of primary importance in algorithmic graph theory due to the fact that many problems being NPhard for general graphs can be solved in polynomial time when restricted to graphs where one of these parameters is bounded. It is known that for any fixed Delta >= 3, all three parameters are unbounded for graphs with vertex degree at most Delta. In this paper, we distinguish representative subclasses of graphs with bounded vertex degree that have bounded band, tree or cliquewidth. Our proofs are constructive and lead to ecient algorithms for a variety of NPhard graph problems when restricted to those classes.
Linear Structure of Bipartite Permutation Graphs and the Longest Path Problem
"... The class of bipartite permutation graphs is the intersection of two well known graph classes: bipartite graphs and permutation graphs. A complete bipartite decomposition of a bipartite permutation graph is proposed in this note. The decomposition gives a linear structure of bipartite permutation gr ..."
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Cited by 6 (0 self)
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The class of bipartite permutation graphs is the intersection of two well known graph classes: bipartite graphs and permutation graphs. A complete bipartite decomposition of a bipartite permutation graph is proposed in this note. The decomposition gives a linear structure of bipartite permutation graphs, and it can be obtained in O(n) time, where n is the number of vertices. As an application of the decomposition, we show an O(n) time and space algorithm for finding a longest path in a bipartite permutation graph.
Cliquewidth and the speed of hereditary properties
"... In this paper, we study the relationship between the number of nvertex graphs in a hereditary class X, also known as the speed of the class X, and boundedness of the cliquewidth in this class. We show that if the speed of X is faster than n!c n for any c, then the cliquewidth of graphs in X is un ..."
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Cited by 4 (3 self)
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In this paper, we study the relationship between the number of nvertex graphs in a hereditary class X, also known as the speed of the class X, and boundedness of the cliquewidth in this class. We show that if the speed of X is faster than n!c n for any c, then the cliquewidth of graphs in X is unbounded, while if the speed does not exceed the Bell number Bn, then the cliquewidth is bounded by a constant. The situation in the range between these two extremes is more complicated. This area contains both classes of bounded and unbounded cliquewidth. Moreover, we show that classes of graphs of unbounded cliquewidth may have slower speed than classes where the cliquewidth is bounded. Keywords: Cliquewidth; Hereditary class of graphs; Speed of hereditary classes 1
Minimal universal bipartite graphs
"... A graph U is (induced)universal for a class of graphs X if every member of X is contained in U as an induced subgraph. We study the problem of nding a universal graph with minimum number of vertices for various classes of bipartite graphs: exponential classes of bipartite (and general) graphs, bipa ..."
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Cited by 3 (2 self)
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A graph U is (induced)universal for a class of graphs X if every member of X is contained in U as an induced subgraph. We study the problem of nding a universal graph with minimum number of vertices for various classes of bipartite graphs: exponential classes of bipartite (and general) graphs, bipartite chain graphs, bipartite permutation graphs, and general bipartite graphs. For exponential classes and general bipartite graphs we present a construction which is asymptotically optimal while for the other classes our solutions are optimal in order. 1
Tractabilities and Intractabilities on Geometric Intersection Graphs
 ALGORITHMS
, 2013
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Cliquewidth for fourvertex forbidden subgraphs
 Theory Comput. Syst
, 2006
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Fast FPT algorithms for vertex subset and vertex partitioning problems using neighborhood unions
, 2009
"... We introduce the graph parameter booleanwidth, related to the number of different unions of neighborhoods across a cut of a graph. Booleanwidth is similar to rankwidth, which is related to the number of GF[2]sums (1+1=0) of neighborhoods instead of the booleansums (1+1=1) used for booleanwidt ..."
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Cited by 2 (2 self)
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We introduce the graph parameter booleanwidth, related to the number of different unions of neighborhoods across a cut of a graph. Booleanwidth is similar to rankwidth, which is related to the number of GF[2]sums (1+1=0) of neighborhoods instead of the booleansums (1+1=1) used for booleanwidth. We give algorithms for a large class of NPhard vertex subset and vertex partitioning problems that are FPT when parameterized by either booleanwidth, rankwidth or cliquewidth, with runtime single exponential in either parameter if given the pertinent optimal decomposition. To compare booleanwidth versus rankwidth or cliquewidth, we first show that for any graph, the square root of its booleanwidth is never more than its rankwidth. Next, we exhibit a class of graphs, the Hsugrids, for which we can solve NPhard problems in polynomial time, if we use the right parameter. An n × n 10 Hsugrid on
Solving MaxSAT and #SAT on structured CNF formulas
 In C. Sinz & U. Egly (Eds.), SAT
, 2014
"... In this paper we propose a structural parameter of CNF formulas and use it to identify instances of weighted MaxSAT and #SAT that can be solved in polynomial time. Given a CNF formula we say that a set of clauses is precisely satisfiable if there is some complete assignment satisfying these clauses ..."
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Cited by 2 (1 self)
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In this paper we propose a structural parameter of CNF formulas and use it to identify instances of weighted MaxSAT and #SAT that can be solved in polynomial time. Given a CNF formula we say that a set of clauses is precisely satisfiable if there is some complete assignment satisfying these clauses only. Let the psvalue of the formula be the number of precisely satisfiable sets of clauses. Applying the notion of branch decompositions to CNF formulas and using psvalue as cut function, we define the pswidth of a formula. For a formula given with a decomposition of polynomial pswidth we show dynamic programming algorithms solving weighted MaxSAT and #SAT in polynomial time. Combining with results of ’Belmonte and Vatshelle, Graph classes with structured neighborhoods and algorithmic applications, Theor. Comput. Sci. 511: 5465 (2013) ’ we get polynomialtime algorithms solving weighted MaxSAT and #SAT for some classes of structured CNF formulas. For example, we get O(m2(m + n)s) algorithms for formulas F of m clauses and n variables and size s, if F has a linear ordering of the variables and clauses such that for any variable x occurring in clause C, if x appears before C then any variable between them also occurs in C, and if C appears before x then x occurs also in any clause between them. Note that the class of incidence graphs of such formulas do not have bounded cliquewidth. 1