Layout of Graphs with Bounded Tree-Width

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 $\mathbb{Z}^3$ 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 $G$ is closely related to the queue-number of $G$. In particular, if $G$ is an $n$-vertex member of a proper minor-closed family of graphs (such as a planar graph), then $G$ has a $O(1)\times O(1)\times O(n)$ drawing if and only if $G$ 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 $k$-trees with applications in graph layout (WG '03), LNCS 2880:205-217, 200

Simple and Efficient Bilayer Cross Counting

We consider the problem of counting the interior edge crossings when a bipartite graph G=(V,E) with node set V and edge set E is drawn such that the nodes of the two shores of the bipartition are on two parallel lines and the edges are straight lines. The efficient solution of this problem is important in layered graph drawing.Our main observation is that it can be reduced to counting the inversions of a certain sequence. This leads to an O(|E|+|C|) algorithm, where C denotes the set of pairwise interior edge crossings, as well as to a simple O(|E|log|V_{m small}|) algorithm, where V_{m small} is the smaller cardinality node set in the bipartition of the node set |V| of the graph. We present the algorithms and the results of computational experiments with these and other algorithms on a large collection of instances

Physics at the LHC Run-2 and Beyond

These lecture notes discuss methods, recent results and future prospects in proton-proton physics at the Large Hadron Collider.Comment: Lecture notes from the 2016 European School of High-Energy Physics, 15-28 June 2016, Skeikampen, Norway (61 pages, 56 figures

Urban identity through quantifiable spatial attributes: Coherence and dispersion of local identity through the comparative analysis of building block plans

The present analysis investigates whether and to what degree quantifiable spatial attributes, as expressed in plan representations, can capture elements related to the experience of spatial identity. Spatial identity is viewed as a constantly rearranging system of relations between discrete singularities. It is proposed that the structure of this system is perceived, inter alia, through its reflection in patterns of variable associations amongst constant spatial features. The examination of such patterns could thus reveal aspects of spatial identity in terms of degrees of differentiation and identification between discrete spatial unities. By combining different methods of shape and spatial analysis it is attempted to quantify spatial attributes, predominantly derived from plans, in order to illustrate patterns of interrelations between spaces through an objective automated process. Variability of methods aims at multileveled spatial descriptions, based on features related to scalar, geometrical and topological attributes of plans. The analysis focuses on the scale of the urban block as the basic modular unit for the formation of urban configurations and the issue of spatial identity is perceived through consistency and differentiation within and amongst urban neighbourhoods. The abstract representation of spatial units enables the investigation of the structure of relations, from which urban identity emerges, based on generic spatial attributes, detached from specific expressions of architectural style

Stacks, Queues and Tracks: Layouts of Graph Subdivisions

A \emphk-stack layout (respectively, \emphk-queuelayout) of a graph consists of a total order of the vertices, and a partition of the edges into k sets of non-crossing (non-nested) edges with respect to the vertex ordering. A \emphk-track layout of a graph consists of a vertex k-colouring, and a total order of each vertex colour class, such that between each pair of colour classes no two edges cross. The \emphstack-number (respectively, \emphqueue-number, \emphtrack-number) of a graph G, denoted by sn(G) (qn(G), tn(G)), is the minimum k such that G has a k-stack (k-queue, k-track) layout.\par This paper studies stack, queue, and track layouts of graph subdivisions. It is known that every graph has a 3-stack subdivision. The best known upper bound on the number of division vertices per edge in a 3-stack subdivision of an n-vertex graph G is improved from O(log n) to O(log min\sn(G),qn(G)\). This result reduces the question of whether queue-number is bounded by stack-number to whether 3-stack graphs have bounded queue number.\par It is proved that every graph has a 2-queue subdivision, a 4-track subdivision, and a mixed 1-stack 1-queue subdivision. All these values are optimal for every non-planar graph. In addition, we characterise those graphs with k-stack, k-queue, and k-track subdivisions, for all values of k. The number of division vertices per edge in the case of 2-queue and 4-track subdivisions, namely O(log qn(G)), is optimal to within a constant factor, for every graph G. \par Applications to 3D polyline grid drawings are presented. For example, it is proved that every graph G has a 3D polyline grid drawing with the vertices on a rectangular prism, and with O(log qn(G)) bends per edge. Finally, we establish a tight relationship between queue layouts and so-called 2-track thickness of bipartite graphs. \pa

Modelling discomfort: How do drivers feel when cyclists cross their path?

Introduction: Even as worldwide interest in bicycling continues to grow, cyclists constitute a large part of road fatalities. A major part of the fatalities occurs when cyclists cross a vehicle path. Active safety systems and automated driving systems may already account for these interactions in their control algorithms. However, the driver behaviour models that these systems use may not be optimal in terms of driver acceptance. If the systems could estimate driver discomfort, their acceptance might be improved.Method: This study investigated the degree of discomfort experienced by drivers when cyclists crossed their travel path. Participants were instructed to drive through an intersection in a fixed-base simulator or on a test track, following the same experimental protocol. The effects of demographic variables (age, gender, driving frequency, and yearly mileage), controlled variables (car speed, bicycle speed, and bicycle-car configuration), and a visual cue (car’s time-to-arrival at the intersection when the bicycle appears; TTAvis) on self-reported discomfort were analysed using cumulative link mixed models (CLMM).Results: Results showed that demographic variables had a significant effect on the discomfort felt by drivers—and could explain the variability observed between drivers. Across both experimental environments, the controlled variables were shown to significantly influence discomfort. TTAvis was shown to have a significant effect on discomfort as well; the closer to zero TTAvis was (i.e., the more critical the situation), the more likely the driver red great discomfort. The prediction accuracies of the CLMM with controlled variables and the CLMM with the visual cue were similar, with an average accuracy between 40 and 50%. Surprise trials in the simulator experiment, in which the bicycle appeared unexpectedly, improved the prediction accuracy of the models, more notably the CLMM including TTAvis. Conclusions: The results suggest that the discomfort was mainly driven by the visual cue rather than the deceleration cues. Thus, it is suggested that an algorithm that estimates driver discomfort be included in active safety systems and autonomous driving systems. The CLMM including TTAvis was presented as a potential candidate to serve this purpose

Kreisplanarität von Level-Graphen

In this dissertation we generalise the notion of level planar graphs in two directions: track planarity and radial planarity. Our main results are linear time algorithms both for the planarity test and for the computation of an embedding, and thus a drawing. Our algorithms use and generalise PQ-trees, which are a data structure for efficient planarity tests.In dieser Arbeit wird der Begriff Level-Planarität von Graphen auf zwei Arten erweitert: Spur-Planarität und radiale Level-Planarität. Die Hauptergebnisse sind Linearzeitalgorithmen zum Testen dieser Arten von Planarität und zur Erstellung einer entsprechenden Einbettung und somit einer Zeichnung. Die Algorithmen verwenden und generalisieren PQ-Bäume, eine bei effizienten Planaritätstests verwendete Datenstruktur