The existing literature gives ecient algorithms for mapping trees or less restrictively outerplanar graphs on a given set of points in a plane, so that the edges are drawn planar and as straight lines. We relax the latter requirement and allow very few bends on each edge while considering general plane graphs. Our results show two algorithms for mapping four-connected plane graphs with at most one bend per edge and for mapping general plane graphs with at most two bends per edge. Furthermore we give a point set, where for arbitrary plane graphs it is NP-complete to decide whether there is an mapping such that each edge has at most one bend.

We present a survey of recent methods for creating piecewise linear mappings between triangulations in 3D and simpler domains such as planar regions, simplicial complexes, and spheres. We also discuss emerging tools such as global parameterization, inter-surface mapping, and parameterization with constraints. We start by describing the wide range of applications where parameterization tools have been used in recent years. We then briefly review the pertinent mathematical background and terminology, before proceeding to survey the existing parameterization techniques. Our survey summarizes the main ideas of each technique and discusses its main properties, comparing it to other methods available. Thus it aims to provide guidance to researchers and developers when assessing the suitability of different methods for various applications. This survey focuses on the practical aspects of the methods available, such as time complexity and robustness and shows multiple examples of parameterizations generated using different methods, allowing the reader to visually evaluate and compare the results.

We consider several variations of the simultaneous embedding problem for planar graphs. We begin with a simple proof that not all pairs of planar graphs have simultaneous geometric embedding. However, using bends, pairs of planar graphs can be simultaneously embedded on the O(n 2) × O(n 2) grid, with at most three bends per edge, where n is the number of vertices. The O(n) time algorithm guarantees that two corresponding vertices in the graphs are mapped to the same location in the final drawing and that both the drawings are crossing-free. The special case when both input graphs are trees has several applications, such as contour tree simplification and evolutionary biology. We show that if both the input graphs are are trees, only one bend per edge is required. The O(n) time algorithm guarantees that both drawings are crossings-free, corresponding tree vertices are mapped to the same locations, and all vertices (and bends) are on the O(n 2) × O(n 2) grid (O(n 3) × O(n 3) grid). For the special case when one of the graphs is a tree and the other is a path we can find simultaneous embedding with fixed-edges. That is, we can guarantee that corresponding vertices are mapped to the same locations and that corresponding edges are drawn the same way. We describe an O(n) time algorithm for simultaneous embedding with fixededges for tree-path pairs with at most one bend per tree-edge and no bends along path edges, such that all vertices (and bends) are on the O(n) × O(n 2) grid, (O(n 2) × O(n 3) grid).

Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and inter-surface mapping, and demonstrates a variety of parameterization applications.

Traditionally, graph drawing algorithms represent vertices as circles and edges as curves connecting the vertices. We introduce the problem of drawing with “fat ” edges, i.e., with edges of variable thickness. The thickness of an edge is often used as a visualization cue, to indicate importance, or to convey some additional information. We present a model for drawing with fat edges and a corresponding polynomial time algorithm that uses the model. We focus on a restricted class of graphs that occur in VLSI wire routing and show how to extend the algorithm to general planar graphs. We show how to convert an arbitrary wire routing into a homotopically equivalent routing that maximizes the distance between any two wires. Among such, we obtain the routing with minimum total wire length. A homotopically equivalent routing that maximizes the distance between any two wires yields a graph drawing which maximizes edge thickness. Finally, our algorithm also allows for different edge weights, that is, the requirement for unit wire thickness can be removed.

Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215–221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.

Abstract. The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004] asked whether every graph of bounded maximum degree has bounded geometric thickness. We answer this question in the negative, by proving that there exists ∆-regular graphs with arbitrarily large geometric thickness. In particular, for all ∆ ≥ 9 and for all large n, there exists a ∆-regular graph with geometric thickness at least c √ ∆n 1/2−4/∆−ǫ. Analogous results concerning graph drawings with few edge slopes are also presented, thus solving open problems by Dujmović et al. [Really straight graph drawings. In Proc. 12th

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Abstract. A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the Euclidean plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same simple curve in the simultaneous drawing. Determining in polynomial time which pairs of graphs share a simultaneous embedding with fixed edges (SEFE) has been open. We give a necessary and sufficient condition for whether a SEFE exists for pairs of graphs whose union is homeomorphic to K5 or K3,3. This allows us to characterize the class of planar graphs that always have a SEFE with any other planar graph. We also characterize the class of biconnected outerplanar graphs that always have a SEFE with any other outerplanar graph. In both cases, we provide efficient algorithms to compute a SEFE. Finally, we provide a linear-time decision algorithm for deciding whether a pair of biconnected outerplanar graphs has a SEFE. 1

We consider planar drawings of trees that must satisfy constraints on the angles between edges incident to a common vertex and on the distances between adjacent vertices. These requirements arise naturally in many applications such as drawing phylogenetic trees or route maps. For straight-line drawings, either class of constraints is always realizable, whereas their combination is not in general. We show that straight-line realizability can be tested in linear time, and give an algorithm that produces drawing satisfying both groups of constraints together in a model where edges are represented as polylines with at most two bends per edge or as continuously differentiable curves.