A graph consists of a set and a binary relation on that set. Each element of the set is a node of the graph, while each element of the binary relation is an edge of the graph that encodes a relationship between two nodes. Graph are pervasive in many areas of science, engineering, and the social sciences: servers on the Internet are connected, proteins interact in large biological systems, social networks encode the relationships between people, and functions call each other in a program. In these domains, the graphs can become very large, consisting of hundreds of thousands of nodes and millions of edges. Graph drawing approaches endeavour to place these nodes in two or three-dimensional space with the intention of fostering an understanding of the binary relation by a human being examining the image. However, many of these approaches to drawing do not exploit higher-level structures in the graph beyond the nodes and edges. Frequently, these structures can be exploited for drawing. As an example, consider a large computer network where nodes are servers and edges are connections between those servers. If a user would like understand how servers at UBC connect to the rest of the network, a drawing that accentuates the set of nodes representing those servers may be more helpful than an approach where all nodes are drawn in the same way. In a feature-based approach, features are subgraphs exploited for the purposes of drawing. We endeavour to depict not only the binary relation, but the high-level relationships between features. This thesis extensively explores a feature-based approach to graph vi sualization and demonstrates the viability of tools that aid in the visual ization of large graphs. Our contributions lie in presenting and evaluating novel techniques and algorithms for graph visualization. We implement five systems in order to empirically evaluate these techniques and algorithms, comparing them to previous approaches.

A graph consists of a set and a binary relation on that set. Each element of the set is a node of the graph, while each element of the binary relation is an edge of the graph that encodes a relationship between two nodes. Graph are pervasive in many areas of science, engineering, and the social sciences: servers on the Internet are connected, proteins interact in large biological systems, social networks encode the relationships between people, and functions call each other in a program. In these domains, the graphs can become very large, consisting of hundreds of thousands of nodes and millions of edges. Graph drawing approaches endeavour to place these nodes in two or three-dimensional space with the intention of fostering an understanding of the binary relation by a human being examining the image. However, many of these approaches to drawing do not exploit higher-level structures in the graph beyond the nodes and edges. Frequently, these structures can be exploited for drawing. As an example, consider a large computer network where nodes are servers and edges are connections between those servers. If a user would like understand how servers at UBC connect to the rest of the network, a drawing that accentuates the set of nodes representing those servers may be more helpful than an approach where all nodes are drawn in the same way. In a feature-based approach, features are subgraphs exploited for the purposes of drawing. We endeavour to depict not only the binary relation, but the high-level relationships between features. This thesis extensively explores a feature-based approach to graph vi sualization and demonstrates the viability of tools that aid in the visual ization of large graphs. Our contributions lie in presenting and evaluating novel techniques and algorithms for graph visualization. We implement five systems in order to empirically evaluate these techniques and algorithms, comparing them to previous approaches.

A point-set embedding of a planar graph G with n vertices on a set S of n points is a planar straight-line drawing of G, where each vertex of G is mapped to a distinct point of S. We prove that the point-set embeddability problem is NP-complete for 3-connected planar graphs, answering a question of Cabello [20]. We give an O(nlog^3n)-time algorithm for testing point-set embeddability of plane 3-trees, improving the algorithm of Moosa and Rahman [60]. We prove that no set of 24 points can support all planar 3-trees with 24 vertices, partially answering a question of Kobourov [55]. We compute 2-bend point-set embeddings of plane 3-trees in O(W^2) area, where W is the length of longest edge of the bounding box of S. Finally, we design algorithms for testing convex point-set embeddability of klee graphs and arbitrary planar graphs.

A point-set embedding of a planar graph G with n vertices on a set S of n points is a planar straight-line drawing of G, where each vertex of G is mapped to a distinct point of S. We prove that the point-set embeddability problem is NP-complete for 3-connected planar graphs, answering a question of Cabello [20]. We give an O(nlog^3n)-time algorithm for testing point-set embeddability of plane 3-trees, improving the algorithm of Moosa and Rahman [60]. We prove that no set of 24 points can support all planar 3-trees with 24 vertices, partially answering a question of Kobourov [55]. We compute 2-bend point-set embeddings of plane 3-trees in O(W^2) area, where W is the length of longest edge of the bounding box of S. Finally, we design algorithms for testing convex point-set embeddability of klee graphs and arbitrary planar graphs.

A graph consists of a set and a binary relation on that set. Each element of the set is a node of the graph, while each element of the binary relation is an edge of the graph that encodes a relationship between two nodes. Graph are pervasive in many areas of science, engineering, and the social sciences: servers on the Internet are connected, proteins interact in large biological systems, social networks encode the relationships between people, and functions call each other in a program. In these domains, the graphs can become very large, consisting of hundreds of thousands of nodes and millions of edges. Graph drawing approaches endeavour to place these nodes in two or three-dimensional space with the intention of fostering an understanding of the binary relation by a human being examining the image. However, many of these approaches to drawing do not exploit higher-level structures in the graph beyond the nodes and edges. Frequently, these structures can be exploited for drawing. As an example, consider a large computer network where nodes are servers and edges are connections between those servers. If a user would like understand how servers at UBC connect to the rest of the network, a drawing that accentuates the set of nodes representing those servers may be more helpful than an approach where all nodes are drawn in the same way. In a feature-based approach, features are subgraphs exploited for the purposes of drawing. We endeavour to depict not only the binary relation, but the high-level relationships between features. This thesis extensively explores a feature-based approach to graph vi sualization and demonstrates the viability of tools that aid in the visual ization of large graphs. Our contributions lie in presenting and evaluating novel techniques and algorithms for graph visualization. We implement five systems in order to empirically evaluate these techniques and algorithms, comparing them to previous approaches. / Science, Faculty of / Computer Science, Department of / Graduate

A pair of straight-line drawings of a graph is called parallel if, for every edge of the graph, the line segment that represents the edge in one drawing is parallel with the line segment that represents the edge in the other drawing. We study the problem of morphing between pairs of parallel planar drawings of a graph, keeping all intermediate drawings planar and parallel with the source and target drawings. We call such a morph a parallel morph. Parallel morphs have application to graph visualization. The problem of deciding whether two parallel drawings in the plane admit a parallel morph turns out to be NP-hard in general. However, for some restricted classes of graphs and drawings, we can efficiently decide parallel morphability. Our main positive result is that every pair of parallel simple orthogonal drawings in the plane admits a parallel morph. We give an efficient algorithm that computes such a morph. The number of steps required in a morph produced by our algorithm is linear in the complexity of the graph, where a step involves moving each vertex along a straight line at constant speed. We prove that this upper bound on the number of steps is within a constant factor of the worst-case lower bound. We explore the related problem of computing a parallel morph where edges are required to change length monotonically, i.e. to be either non-increasing or non-decreasing in length. Although parallel orthogonally-convex polygons always admit a monotone parallel morph, deciding morphability under these constraints is NP-hard, even for orthogonal polygons. We also begin a study of parallel morphing in higher dimensions. Parallel drawings of trees in any dimension always admit a parallel morph. This is not so for parallel drawings of cycles in 3-space, even if orthogonal. Similarly, not all pairs of parallel orthogonal polyhedra admit a parallel morph, even if they are topological spheres. In fact, deciding parallel morphability turns out to be PSPACE-hard for both parallel orthogonal polyhedra, and parallel orthogonal drawings in 3-space.

A pair of straight-line drawings of a graph is called parallel if, for every edge of the graph, the line segment that represents the edge in one drawing is parallel with the line segment that represents the edge in the other drawing. We study the problem of morphing between pairs of parallel planar drawings of a graph, keeping all intermediate drawings planar and parallel with the source and target drawings. We call such a morph a parallel morph. Parallel morphs have application to graph visualization. The problem of deciding whether two parallel drawings in the plane admit a parallel morph turns out to be NP-hard in general. However, for some restricted classes of graphs and drawings, we can efficiently decide parallel morphability. Our main positive result is that every pair of parallel simple orthogonal drawings in the plane admits a parallel morph. We give an efficient algorithm that computes such a morph. The number of steps required in a morph produced by our algorithm is linear in the complexity of the graph, where a step involves moving each vertex along a straight line at constant speed. We prove that this upper bound on the number of steps is within a constant factor of the worst-case lower bound. We explore the related problem of computing a parallel morph where edges are required to change length monotonically, i.e. to be either non-increasing or non-decreasing in length. Although parallel orthogonally-convex polygons always admit a monotone parallel morph, deciding morphability under these constraints is NP-hard, even for orthogonal polygons. We also begin a study of parallel morphing in higher dimensions. Parallel drawings of trees in any dimension always admit a parallel morph. This is not so for parallel drawings of cycles in 3-space, even if orthogonal. Similarly, not all pairs of parallel orthogonal polyhedra admit a parallel morph, even if they are topological spheres. In fact, deciding parallel morphability turns out to be PSPACE-hard for both parallel orthogonal polyhedra, and parallel orthogonal drawings in 3-space.

A straight line drawing of a graph is an open weak rectangle-of-influence (RI) drawing, if there is no vertex in the relative interior of the axis parallel rectangle induced by the end points of each edge. Despite recent interest of the graph drawing community in rectangle-of-influence drawings, no algorithm is known to test whether a graph has a planar open weak RI-drawing, not even for inner triangulated graphs. In this thesis, we have two major contributions. First we study open weak RI-drawings of plane graphs that must have a non-aligned frame, i.e., the graph obtained from removing the interior of every filled triangle is drawn such that no two vertices have the same coordinate. We introduce a new way to assign labels to angles, i.e., instances of vertices on faces. Using this labeling, we provide necessary and sufficient conditions characterizing those plane graphs that have open weak RI-drawings with non-aligned frame. We also give a polynomial algorithm to construct such a drawing if one exists. Our second major result is a negative result: deciding if a planar graph (i.e., one where we can choose the planar embedding) has an open weak RI-drawing is NP-complete. NP-completeness holds even for open weak RI-drawings with non-aligned frames.

A straight line drawing of a graph is an open weak rectangle-of-influence (RI) drawing, if there is no vertex in the relative interior of the axis parallel rectangle induced by the end points of each edge. Despite recent interest of the graph drawing community in rectangle-of-influence drawings, no algorithm is known to test whether a graph has a planar open weak RI-drawing, not even for inner triangulated graphs. In this thesis, we have two major contributions. First we study open weak RI-drawings of plane graphs that must have a non-aligned frame, i.e., the graph obtained from removing the interior of every filled triangle is drawn such that no two vertices have the same coordinate. We introduce a new way to assign labels to angles, i.e., instances of vertices on faces. Using this labeling, we provide necessary and sufficient conditions characterizing those plane graphs that have open weak RI-drawings with non-aligned frame. We also give a polynomial algorithm to construct such a drawing if one exists. Our second major result is a negative result: deciding if a planar graph (i.e., one where we can choose the planar embedding) has an open weak RI-drawing is NP-complete. NP-completeness holds even for open weak RI-drawings with non-aligned frames.

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