34 research outputs found

    Stack and Queue Layouts of Posets

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    The stacknumber (queuenumber) of a poset is defined as the stacknumber (queuenumber) of its Hasse diagram viewed as a directed acyclic graph. Upper bounds on the queuenumber of a poset are derived in terms of its jumpnumber, its length, its width, and the queuenumber of its covering graph. A lower bound of is shown for the queuenumber of the class of planar posets. The queuenumber of a planar poset is shown to be within a small constant factor of its width. The stacknumber of posets with planar covering graphs is shown to be . These results exhibit sharp differences between the stacknumber and queuenumber of posets as well as between the stacknumber (queuenumber) of a poset and the stacknumber (queuenumber) of its covering graph

    The Queue-Number of Posets of Bounded Width or Height

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    Heath and Pemmaraju conjectured that the queue-number of a poset is bounded by its width and if the poset is planar then also by its height. We show that there are planar posets whose queue-number is larger than their height, refuting the second conjecture. On the other hand, we show that any poset of width 22 has queue-number at most 22, thus confirming the first conjecture in the first non-trivial case. Moreover, we improve the previously best known bounds and show that planar posets of width ww have queue-number at most 3w23w-2 while any planar poset with 00 and 11 has queue-number at most its width.Comment: 14 pages, 10 figures, Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

    Layout of Graphs with Bounded Tree-Width

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    A \emph{queue layout} of a graph consists of a total order of the vertices, and a partition of the edges into \emph{queues}, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its \emph{queue-number}. A \emph{three-dimensional (straight-line grid) drawing} of a graph represents the vertices by points in Z3\mathbb{Z}^3 and the edges by non-crossing line-segments. This paper contributes three main results: (1) It is proved that the minimum volume of a certain type of three-dimensional drawing of a graph GG is closely related to the queue-number of GG. In particular, if GG is an nn-vertex member of a proper minor-closed family of graphs (such as a planar graph), then GG has a O(1)×O(1)×O(n)O(1)\times O(1)\times O(n) drawing if and only if GG has O(1) queue-number. (2) It is proved that queue-number is bounded by tree-width, 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 in the theory of queue layouts. (3) It is proved that graphs of bounded tree-width have three-dimensional drawings with O(n) volume. This is the most general family of graphs known to admit three-dimensional drawings with O(n) volume. The proofs depend upon our results regarding \emph{track layouts} and \emph{tree-partitions} of graphs, which may be of independent interest.Comment: This is a revised version of a journal paper submitted in October 2002. This paper incorporates the following conference papers: (1) Dujmovic', Morin & Wood. Path-width and three-dimensional straight-line grid drawings of graphs (GD'02), LNCS 2528:42-53, Springer, 2002. (2) Wood. Queue layouts, tree-width, and three-dimensional graph drawing (FSTTCS'02), LNCS 2556:348--359, Springer, 2002. (3) Dujmovic' & Wood. Tree-partitions of kk-trees with applications in graph layout (WG '03), LNCS 2880:205-217, 200

    On Families of Planar DAGs with Constant Stack Number

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    A kk-stack layout (or kk-page book embedding) of a graph consists of a total order of the vertices, and a partition of the edges into kk sets of non-crossing edges with respect to the vertex order. The stack number of a graph is the minimum kk such that it admits a kk-stack layout. In this paper we study a long-standing problem regarding the stack number of planar directed acyclic graphs (DAGs), for which the vertex order has to respect the orientation of the edges. We investigate upper and lower bounds on the stack number of several families of planar graphs: We prove constant upper bounds on the stack number of single-source and monotone outerplanar DAGs and of outerpath DAGs, and improve the constant upper bound for upward planar 3-trees. Further, we provide computer-aided lower bounds for upward (outer-) planar DAGs

    Characterisations and Examples of Graph Classes with Bounded Expansion

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    Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Ne\v{s}et\v{r}il and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several linear-time algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties. In this paper we establish two new characterisations of bounded expansion classes, one in terms of so-called topological parameters, the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with Erd\"os-R\'enyi model of random graphs with constant average degree. In particular, we prove that for every fixed d>0d>0, there exists a class with bounded expansion, such that a random graph of order nn and edge probability d/nd/n asymptotically almost surely belongs to the class. We then present several new examples of classes with bounded expansion that do not exclude some topological minor, and appear naturally in the context of graph drawing or graph colouring. In particular, we prove that the following classes have bounded expansion: graphs that can be drawn in the plane with a bounded number of crossings per edge, graphs with bounded stack number, graphs with bounded queue number, and graphs with bounded non-repetitive chromatic number. We also prove that graphs with `linear' crossing number are contained in a topologically-closed class, while graphs with bounded crossing number are contained in a minor-closed class

    Directed Acyclic Outerplanar Graphs Have Constant Stack Number

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    The stack number of a directed acyclic graph GG is the minimum kk for which there is a topological ordering of GG and a kk-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological ordering. We prove that the stack number of directed acyclic outerplanar graphs is bounded by a constant, which gives a positive answer to a conjecture by Heath, Pemmaraju and Trenk [SIAM J. Computing, 1999]. As an immediate consequence, this shows that all upward outerplanar graphs have constant stack number, answering a question by Bhore et al. [GD 2021] and thereby making significant progress towards the problem for general upward planar graphs originating from Nowakowski and Parker [Order, 1989]. As our main tool we develop the novel technique of directed HH-partitions, which might be of independent interest. We complement the bounded stack number for directed acyclic outerplanar graphs by constructing a family of directed acyclic 2-trees that have unbounded stack number, thereby refuting a conjecture by N\"ollenburg and Pupyrev [arXiv:2107.13658, 2021]

    Crossing-Free Acyclic Hamiltonian Path Completion for Planar st-Digraphs

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    In this paper we study the problem of existence of a crossing-free acyclic hamiltonian path completion (for short, HP-completion) set for embedded upward planar digraphs. In the context of book embeddings, this question becomes: given an embedded upward planar digraph GG, determine whether there exists an upward 2-page book embedding of GG preserving the given planar embedding. Given an embedded stst-digraph GG which has a crossing-free HP-completion set, we show that there always exists a crossing-free HP-completion set with at most two edges per face of GG. For an embedded NN-free upward planar digraph GG, we show that there always exists a crossing-free acyclic HP-completion set for GG which, moreover, can be computed in linear time. For a width-kk embedded planar stst-digraph GG, we show that we can be efficiently test whether GG admits a crossing-free acyclic HP-completion set.Comment: Accepted to ISAAC200

    Combinatorial Structures in Hypercubes

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