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24
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 89 (4 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 54 (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.
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.
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 13 (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
Logical Aspects of CayleyGraphs: The Group Case
 TO APPEAR IN ANNALS OF PURE AND APPLIED LOGIC
"... We prove that a finitely generated group is contextfree whenever its Cayleygraph has a decidable monadic secondorder theory. Hence, by the seminal work of Muller and Schupp, our result gives a logical characterization of contextfree groups and also proves a conjecture of Schupp. To derive this re ..."
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Cited by 12 (3 self)
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We prove that a finitely generated group is contextfree whenever its Cayleygraph has a decidable monadic secondorder theory. Hence, by the seminal work of Muller and Schupp, our result gives a logical characterization of contextfree groups and also proves a conjecture of Schupp. To derive this result, we investigate general graphs and show that a graph of bounded degree with a high degree of symmetry is contextfree whenever its monadic secondorder theory is decidable. Further, it is shown that the word problem of a finitely generated group is decidable if and only if the firstorder theory of its Cayleygraph is decidable.
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 10 (7 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
Isomorphism for Graphs of Bounded Distance Width
"... In this paper, we study the Graph Isomorphism problem on graphs of bounded treewidth, bounded degree or bounded bandwidth. Graph Isomorphism can be solved in polynomial time for graphs of bounded treewidth, pathwidth, or bandwidth, but the exponent depends on the treewidth, pathwidth, or bandwidt ..."
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Cited by 9 (0 self)
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In this paper, we study the Graph Isomorphism problem on graphs of bounded treewidth, bounded degree or bounded bandwidth. Graph Isomorphism can be solved in polynomial time for graphs of bounded treewidth, pathwidth, or bandwidth, but the exponent depends on the treewidth, pathwidth, or bandwidth. Thus, we look for special cases where `fixed parameter tractable' polynomial time algorithms can be established. We introduce some new and natural graph parameters: the (rooted) path distance width, which is a restriction of bandwidth, and the (rooted) tree distance width, which is a restriction of treewidth. We give algorithms that solve Graph Isomorphism in O(n²) time for graphs with bounded rooted path distance width, and in O(n³) time for graphs with bounded rooted tree distance width. Additionally, we show that computing the path distance width of a graph is NPhard, but both path and tree distance width can be computed in O(n k+1 ) time, when they are bounded by a...
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 8 (5 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
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 7 (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.