Cologne Excellence Cluster on Cellular Stress Responses in Aging Associated Diseases
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Graph Simultaneous Embedding Tool, GraphSET
No full textProblems in simultaneous graph drawing involve the layout of several graphs on a shared vertex set. This paper describes a Graph Simultaneous Embedding Tool, GraphSET, designed to allow the investigation of a wide range of embedding problems. GraphSET can be used in the study of several variants of simultaneous embedding including simultaneous geometric embedding, simultaneous embedding with fixed edges and colored simultaneous embedding with the vertex set
partitioned into color classes. The tool has two primary uses: (i) studying theoretical problems in simultaneous graph drawing through the production of examples and counterexamples and (ii) producing layouts of given classes of graphs using built-in implementations of known algorithms. GraphSET along with movies illustrating its utility are available at http://graphset.cs.arizona.edu
Hamiltonian Alternating Paths on Bicolored Double-Chains
No full textWe find arbitrarily large finite sets S of points in general position in the plane with the following property. If the points of S are equitably 2-colored (i.e., the sizes of the two color classes differ by at most one), then there is a polygonal line consisting of straight-line segments with endpoints in S , which is Hamiltonian, non-crossing, and alternating (i.e., each point of S is visited exactly once, every two non-consecutive segments are disjoint, and every segment connects points of different colors). We show that the above property holds for so-called double-chains with each of the two chains containing at least one fifth of all the points. Our proof is constructive and can be turned into a linear-time algorithm. On the other hand, we show that the above property does not hold for double-chains in which one of the chains contains at most ≈ 1/29 of all the points
Generalizing the Shift Method for Rectangular Shaped Vertices with Visibility Constraints
No full textIn this paper we present a generalization of the shift method algorithm to obtain a straight-line grid drawing of a triconnected graph, where vertex representations have a certain specified size. We propose vertex representations having a rectangular shape. Additionally, one may demand maintainance of the criterion of strong visibility, that is, any possible line segment connecting two adjacent vertices cannot cross another vertex’ representation. We prove that the proposed method produces a straight-line grid drawing of a graph in linear time with an area bound, that is only extended by the size of the rectangles, compared to the bound of the original algorithm
Visual Analysis of One-To-Many Matched Graphs
No full textMotivated by applications of social network analysis and of Web search clustering engines, we describe an algorithm and a system for the display and the visual analysis of two graphs G1 and G2 such that each Gi is defined on a different data set with its own primary relationships and there are secondary relationships between the vertices of G1 and those of G2 . Our main goal is to compute a drawing of G1 and G2 that makes clearly visible the relations between the two graphs by avoiding their crossings, and that also takes into account some other important aesthetic requirements like number of bends, area, and aspect ratio. Application examples and experiments on the system performances are also presented
Embeddability Problems for Upward Planar Digraphs
No full textWe study two embedding problems for upward planar digraphs. Both problems arise in the context of drawing sequences of upward planar digraphs having the same set of vertices, where the location of each vertex is to remain the same for all the drawings of the graphs. We develop a method, based on the notion of book embedding, that gives characterization results for embeddability as well as testing and drawing algorithms
A Fully Dynamic Algorithm to Test the Upward Planarity of Single-Source Embedded Digraphs
No full textIn this chapter, we present a dynamic algorithm that checks if a singlesource embedded digraph is upward planar in the presence of edge insertions and edge deletions. Let Gφ be an upward planar single-source embedded digraph and ′ let Gφ ′ be a single-source embedded digraph obtained by updating Gφ . We show ′ that the upward planarity of Gφ ′ can be checked in O(log n) amortized time when the external face is fixed
Cubic Graphs Have Bounded Slope Parameter
No full textWe show that every finite connected graph G with maximum degree three and with at least one vertex of degree smaller than three has a straight-line drawing in the plane satisfying the following conditions. No three vertices are collinear, and a pair of vertices form an edge in G if and only if the segment connecting them is parallel to one of the sides of a previously fixed regular pentagon. It is also proved that every finite graph with maximum degree three permits a straight-line drawing with the above properties using only at most seven different edge slopes
Unimaximal Sequences of Pairs in Rectangle Visibility Drawing
No full textWe study the existence of unimaximal subsequences in sequences of pairs of integers, e.g., the subsequences that have exactly one local maximum in each component of the subsequence. We show that 1 every sequence of 12 n2 (n2 − 1) + 1 pairs has a unimaximal subsequence of length n. We prove that this bound is tight. We apply this result to the problem of the largest complete graph with a 3D rectangle visibility representation and improve the upper bound from 55 to 50
The Topology of Bendless Three-Dimensional Orthogonal Graph Drawing
No full textWe define an xyz graph to be a spatial embedding of a 3-regular graph such that the edges at each vertex are mutually perpendicular and no three points lie on an axis-parallel line. We describe an equivalence between xyz graphs and 3-face-colored polyhedral maps, under which bipartiteness of the graph is equivalent to orientability of the map. We show that planar graphs are xyz graphs if and only if they are bipartite, cubic, and three-connected. It is NP-complete to recognize xyz graphs, but we show how to do this in time O(n2n/2 )
An SPQR-Tree Approach to Decide Special Cases of Simultaneous Embedding with Fixed Edges
No full textWe present a linear-time algorithm for solving the simultaneous embedding problem with fixed edges (SEFE) for a planar graph and a pseudoforest (a graph with at most one cycle) by reducing it to the following embedding problem: Given a planar graph G, a cycle C of G, and a partitioning of the remaining vertices of G, does there exist a planar embedding in which the induced subgraph on each vertex partite of G \ C is contained entirely inside or outside C ? For the latter problem, we present an algorithm that is based on SPQR-trees and has linear running time. We also show how we can employ SPQR-trees to decide SEFE for two planar graphs where one graph has at most two cycles and the intersection is a pseudoforest in linear time. These results give rise to our hope that our SPQR-tree approach might eventually lead to a polynomial-time algorithm for deciding the general SEFE problem for two planar graphs