In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A k-stack (resp...

Abstract. A tree-partition 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 k-tree has a tree-partition such that each bag induces a (k − 1)-tree, amongst other properties. Applications of this result to two well-studied models of graph layout are presented. First it is proved that graphs of bounded tree-width 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 tree-width have three-dimensional straight-line grid drawings with linear volume, which represents the largest known class of graphs with such drawings. 1

We give a drawing of Kn in three dimensions in which vertices are placed at integer grid points and edges are drawn crossing-free with at most one bend per edge in a volume bounded by O(n^2.5).

We consider three-dimensional grid-drawings of graphs with at most one bend per edge. Under the additional requirement that the vertices be collinear, we prove that the minimum volume of such a drawing is Θ(cn), where n is the number of vertices and c is the cutwidth of the graph. We then prove that every graph has a three-dimensional grid-drawing with O(n 3 / log 2 n) volume and one bend per edge. The best previous bound was O(n 3).

Abstract. A three-dimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce three-dimensional grid drawings with small bounding box volume. Our first main result is that every n-vertex graph with bounded degeneracy has a three-dimensional grid drawing with O(n 3/2) volume. This is the largest known class of graphs that have such drawings. A three-dimensional grid drawing of a directed acyclic graph (dag) is upward if every arc points up in the z-direction. We prove that every dag has an upward three-dimensional grid drawing with O(n 3) volume, which is tight for the complete dag. The previous best upper bound was O(n 4). Our main result concerning upward drawings is that every c-colourable dag (c constant) has an upward three-dimensional grid drawing with O(n 2) volume. This result matches the bound in the undirected case, and improves the best known bound from O(n 3) for many classes of dags, including planar, series parallel, and outerplanar. Improved bounds are also obtained for tree dags. We prove a strong relationship between upward three-dimensional grid drawings, upward track layouts, and upward queue layouts. Finally, we study upward three-dimensional grid drawings with bends in the edges. 1.

The no-three-in-line problem, introduced by Dudeney in 1917, asks for the maximum number of points in the nn grid with no three points collinear. In 1951, Erdos proved that the answer is (n). We consider the analogous three-dimensional problem, and prove that the maximum number of points in the n n n grid with no three collinear is (n ).

Denote by ∇1(G) the maximum of |E(H)| over all (simple) minors |V (H)| of G obtained by contracting a star forest. We prove that there exists a positive function ǫ such that every graph G of order n has (at least) two clones (that is two vertices with the same neighbours) or an induced matching of size at least ǫ(∇1(G))n and that this set may be found in linear time. More generally, we prove that for every integer k there exists a (very slowly growing) positive function ǫk such that every graph of order n has an involutive automorphism or includes a set of size at least k⌊ǫk(∇⌊k/2⌋(G))n ⌋ inducing ⌊ǫk(∇⌊k/2⌋(G))n ⌋ disjoint paths on k vertices.

The GLuskap system for interactive three-dimensional graph drawing applies techniques of scientific visualization and interactive systems to the construction, display, and analysis of graph drawings. Important features of the system include support for large-screen stereo-graphic 3D display with immersive head-tracking and motion-tracked interactive 3D wand control. A distributed rendering architecture contributes to the portability of the system, with user control performed on a laptop computer without specialized graphics hardware. An interface for implementing graph drawing layout and analysis algorithms in the Python programming language is also provided. This thesis describes comprehensively the work on the system by the author—this work includes the design and implementation of the ma-jor features described above. Further directions for continued development and research in cognitive tools for graph drawing research are also suggested.

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We prove that planar graphs have O(log 4 n) queue number, thus improving upon the previous O ( √ n) upper bound. Consequently, planar graphs admit 3D straight-line crossingfree grid drawings in O(n log c n) volume, for some constant c, thus improving upon the previous O(n 3/2) upper bound. 2 1