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19
Domino treewidth
 DISCRETE MATH. THEOR. COMPUT. SCI
, 1994
"... We consider a special variant of treedecompositions, called domino treedecompositions, and the related notion of domino treewidth. In a domino treedecomposition, each vertex of the graph belongs to at most two nodes of the tree. We prove that for every k, d, there exists a constant ck;d such that ..."
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Cited by 75 (3 self)
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We consider a special variant of treedecompositions, called domino treedecompositions, and the related notion of domino treewidth. In a domino treedecomposition, each vertex of the graph belongs to at most two nodes of the tree. We prove that for every k, d, there exists a constant ck;d such that a graph with treewidth at most k and maximum degree at most d has domino treewidth at most ck;d. The domino treewidth of a tree can be computed in O(n 2 log n) time. There exist polynomial time algorithms that  for fixed k  decide whether a given graph G has domino treewidth at most k. If k is not fixed, this problem is NPcomplete. The domino treewidth problem is hard for the complexity classes W [t] for all t 2 N, and hence the problem for fixed k is unlikely to be solvable in O(n c), where c is a constant, not depending on k.
The Structure of the Models of Decidable Monadic Theories of Graphs
, 1991
"... In this article the structure of the models of decidable (weak) monadic theories of planar graphs is investigated. It is shown that if the (weak) monadic theory of a class K of planar graphs is decidable, then the treewidth in the sense of Robertson and Seymour (1984) of the elements of K is univer ..."
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Cited by 47 (2 self)
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In this article the structure of the models of decidable (weak) monadic theories of planar graphs is investigated. It is shown that if the (weak) monadic theory of a class K of planar graphs is decidable, then the treewidth in the sense of Robertson and Seymour (1984) of the elements of K is universally bounded and there is a class T of trees such that the (weak) monadic theory of K is interpretable in the (weak) monadic theory of T.
Width parameters beyond treewidth and their applications
 Computer Journal
, 2007
"... Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compare ..."
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Cited by 19 (0 self)
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Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compared to trees have been born and studied over the past years. These concepts and parameters have proved to be useful tools in many applications, especially in the design of efficient algorithms. Our presented novel look at the contemporary developments of these ‘width ’ parameters in combinatorial structures delivers—besides traditional treewidth and derived dynamic programming schemes—also a number of other useful parameters like branchwidth, rankwidth (cliquewidth) or hypertreewidth. In this contribution, we demonstrate how ‘width ’ parameters of graphs and generalized structures (such as matroids or hypergraphs), can be used to improve the design of parameterized algorithms and the structural analysis in other applications on an abstract level.
Treepartitions of ktrees with applications in graph layout
 Proc. 29th Workshop on Graph Theoretic Concepts in Computer Science (WG’03
, 2002
"... Abstract. A treepartition of a graph is a partition of its vertices into ‘bags ’ such that contracting each bag into a single vertex gives a forest. It is proved that every ktree has a treepartition such that each bag induces a (k − 1)tree, amongst other properties. Applications of this result t ..."
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Cited by 16 (11 self)
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Abstract. A treepartition of a graph is a partition of its vertices into ‘bags ’ such that contracting each bag into a single vertex gives a forest. It is proved that every ktree has a treepartition such that each bag induces a (k − 1)tree, amongst other properties. Applications of this result to two wellstudied models of graph layout are presented. First it is proved that graphs of bounded treewidth have bounded queuenumber, thus resolving an open problem due to Ganley and Heath [2001] and disproving a conjecture of Pemmaraju [1992]. This result provides renewed hope for the positive resolution of a number of open problems regarding queue layouts. In a related result, it is proved that graphs of bounded treewidth have threedimensional straightline grid drawings with linear volume, which represents the largest known class of graphs with such drawings. 1
Planar decompositions and the crossing number of graphs with an excluded minor
 IN GRAPH DRAWING 2006; LECTURE NOTES IN COMPUTER SCIENCE 4372
, 2007
"... Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar ..."
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Cited by 14 (1 self)
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Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number. In particular, a graph with bounded degree has linear crossing number if and only if it has a planar decomposition with bounded width and linear order. It follows from the above result about planar decompositions that every graph with bounded degree and an excluded minor has linear crossing number. Analogous results are proved for the convex and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded treewidth has linear convex crossing number, and every K3,3minorfree graph with bounded degree has linear rectilinear crossing number.
Queue layouts, treewidth, and threedimensional graph drawing
 Proc. 22nd Foundations of Software Technology and Theoretical Computer Science (FST TCS '02
, 2002
"... Abstract. A threedimensional (straightline grid) drawing of a graph represents the vertices by points in Z 3 and the edges by noncrossing line segments. This research is motivated by the following open problem due to Felsner, Liotta, and Wismath [Graph Drawing ’01, Lecture Notes in Comput. Sci., ..."
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Cited by 12 (6 self)
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Abstract. A threedimensional (straightline grid) drawing of a graph represents the vertices by points in Z 3 and the edges by noncrossing line segments. This research is motivated by the following open problem due to Felsner, Liotta, and Wismath [Graph Drawing ’01, Lecture Notes in Comput. Sci., 2002]: does every nvertex planar graph have a threedimensional drawing with O(n) volume? We prove that this question is almost equivalent to an existing onedimensional graph layout problem. A queue layout consists of a linear order σ of the vertices of a graph, and a partition of the edges into queues, such that no two edges in the same queue are nested with respect to σ. The minimum number of queues in a queue layout of a graph is its queuenumber. Let G be an nvertex member of a proper minorclosed family of graphs (such as a planar graph). We prove that G has a O(1) × O(1) × O(n) drawing if and only if G has O(1) queuenumber. Thus the above question is almost equivalent to an open problem of Heath, Leighton, and Rosenberg [SIAM J. Discrete Math., 1992], who ask whether every planar graph has O(1) queuenumber? We also present partial solutions to an open problem of Ganley and Heath [Discrete Appl. Math., 2001], who ask whether graphs of bounded treewidth have bounded queuenumber? We prove that graphs with bounded pathwidth, or both bounded treewidth and bounded maximum degree, have bounded queuenumber. As a corollary we obtain threedimensional drawings with optimal O(n) volume, for seriesparallel graphs, and graphs with both bounded treewidth and bounded maximum degree. 1
Really straight graph drawings
 Proc. 12th International Symp. on Graph Drawing (GD ’04
, 2004
"... We study straightline drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5n/2 segme ..."
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Cited by 10 (3 self)
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We study straightline drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes. We prove that every cubic 3connected plane graph has a plane drawing with three slopes (and three bends on the outerface). Drawings of nonplanar graphs with few slopes are also considered. For example, interval graphs, cocomparability graphs and ATfree graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with O(log n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the
A note on treepartitionwidth
, 2006
"... Abstract. A treepartition of a graph G is a proper partition of its vertex set into ‘bags’, such that identifying the vertices in each bag produces a forest. The treepartitionwidth of G is the minimum number of vertices in a bag in a treepartition of G. An anonymous referee of the paper by Ding ..."
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Cited by 6 (4 self)
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Abstract. A treepartition of a graph G is a proper partition of its vertex set into ‘bags’, such that identifying the vertices in each bag produces a forest. The treepartitionwidth of G is the minimum number of vertices in a bag in a treepartition of G. An anonymous referee of the paper by Ding and Oporowski [J. Graph Theory, 1995] proved that every graph with treewidth k ≥ 3 and maximum degree ∆ ≥ 1 has treepartitionwidth at most 24k∆. We prove that this bound is within a constant factor of optimal. In particular, for all k ≥ 3 and for all sufficiently large ∆, we construct a graph with treewidth k, maximum degree ∆, and treepartitionwidth at least ( 1 8 upper bound to 5
The maximum subforest problem: Approximation and exact algorithms
 IN PROC. 9TH SYMPOSIUM ON DISCRETE ALGORITHMS (SODA 98)
, 1998
"... We study the maximum subforest problem: Given a tree G and a set of trees H, find a subgraph G ′ of G such that G ′ does not contain a subtree isomorphic to a tree from H, and the number of edges in G ′ is maximum. We give a polynomial time approximation scheme for this problem. We also give an exac ..."
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Cited by 6 (3 self)
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We study the maximum subforest problem: Given a tree G and a set of trees H, find a subgraph G ′ of G such that G ′ does not contain a subtree isomorphic to a tree from H, and the number of edges in G ′ is maximum. We give a polynomial time approximation scheme for this problem. We also give an exact algorithm for this problem whose time complexity is 2 O(k2 / log k) n, where n is the number of vertices in G, and k is the total number of vertices in H.
Notes on nonrepetitive graph colouring
 Electron. J. Combin
, 2008
"... A vertex colouring of a graph is nonrepetitive on paths if there is no path v1, v2,..., v2t such that vi and vt+i receive the same colour for all i = 1, 2,..., t. We determine the maximum density of a graph that admits a kcolouring that is nonrepetitive on paths. We prove that every graph has a sub ..."
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Cited by 5 (4 self)
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A vertex colouring of a graph is nonrepetitive on paths if there is no path v1, v2,..., v2t such that vi and vt+i receive the same colour for all i = 1, 2,..., t. We determine the maximum density of a graph that admits a kcolouring that is nonrepetitive on paths. We prove that every graph has a subdivision that admits a 4colouring that is nonrepetitive on paths. The best previous bound was 5. We also study colourings that are nonrepetitive on walks, and provide a conjecture that would imply that every graph with maximum degree ∆ has a f(∆)colouring that is nonrepetitive on walks. We prove that every graph with treewidth k and maximum degree ∆ has a O(k∆)colouring that is nonrepetitive on paths, and a O(k ∆ 3)colouring that is nonrepetitive on walks. 1