10660638
The Metro Map Layout Problem
We initiate a new problem of automatic metro map layout. In general, a metro map consists of a set of lines which have intersections or overlaps. We define a set of aesthetic criteria for good metro map layouts and present a method to produce such layouts automatically. Our method uses a variation of the spring algorithm with a suitable preprocessing step. The experimental results with real world data sets show that our method produces good metro map layouts quickly.
3107205
Crossing Minimization for Symmetries
AbstractWe consider the problem of drawing a graph with a given symmetry
 such that the number of edge crossings is minimal. We show that this
 problem is NP-hard, even if the order of orbits around the rotation
 center or along the reflection axis is fixed.
 We devise an O(m log m) algorithm
 for computing a crossing minimal drawing if inter-orbit edges may
 not cross orbits, showing in particular that intra-orbit edges do
 not contribute to the NP-hardness of the crossing minimization
 problem for symmetries.
30474248
Crossing Minimization for Symmetries
We consider the problem of drawing a graph with a given symmetry such that the number of edge crossings is minimal. We show that this problem is NP-hard, even if the order of orbits around the rotation center or along the reflection axis is fixed. Nevertheless, there is a linear time algorithm to test planarity and to construct a planar embedding if possible. Finally, we devise an O (m log m) algorithm for computing a crossing minimal drawing if inter-orbit edges may not cross orbits, showing in particular that intra-orbit edges do not contribute to the NP-hardness of the crossing minimization problem for symmetries. From this result, we can derive an O (m log m) crossing minimization algorithm for symmetries with an orbit graph that is a path.
6056384
Linkless symmetric drawings of series parallel digraphs
In this paper, we present a linear time algorithm for constructing linkless drawings of series parallel digraphs with maximum number of symmetries. Linkless drawing in three dimensions is a natural extension to planar drawing in two dimensions. Symmetry is one of the most important aesthetic criteria in graph drawing. More specifically, we present two algorithms: a symmetry finding algorithm which finds maximum number of three dimensional symmetries, and a drawing algorithm which constructs linkless symmetric drawings of series parallel digraphs in three dimensions.
9291771
Drawing Trees Symmetrically in Three Dimensions
AbstractSymmetric graph drawing enables a clear understanding of the structure of the graph. Previous work on symmetric graph drawing has focused on two dimensions. Symmetry in three dimensions is much richer than that of two dimensions. This is the first paper to extend symmetric graph drawing into three dimensions. More specifically, the paper investigates the problem of drawing trees symmetrically in three dimensions. First, we suggest a model for drawing trees symmetrically in three dimensions. Based on this model, we present a linear time algorithm for finding the maximum number of three-dimensional symmetries in trees. We also present a three-dimensional symmetric drawing algorithm for trees.

341144
A Linear Time Algorithm for Constructing Maximally Symmetric Straight Line Drawings of Triconnected Planar Graphs
Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals structure in the graph. To draw graphs symmetrically, we employ two steps. The first step is to find appropriate automorphisms. The second step is to draw the graph to display the automorphisms. Our aim in this paper is to construct maximally symmetric straight line drawings of triconnected planar graphs in linear time. Previously known algorithms run in quadratic time. We show that an algorithm of Fontet can be used to find an embedding in the plane with the maximum number of symmetries, and present a new algorithm for finding a straight line drawing that achieves that maximum. Both algorithms run in linear time.
7349909
Reconstructing a formal security model
Abstract Role-based access control (RBAC) is a flexible approach to access control, which has generated great interest in the security community. The principal motivation behind RBAC is to simplify the complexity of administrative tasks. Several formal models of RBAC have been introduced. However, there are a few works specifying RBAC in a way which system developers or software engineers can easily understand and adopt to develop role-based systems. And there still exists a demand to have a practical representation of well-known access control models for system developers who work on secure system development. In this paper we represent a well-known RBAC model with software engineering tools such as Unified Modeling Language (UML) and Object Constraints Language (OCL) to reduce a gap between security models and system developments. The UML is a general-purpose visual modeling language in which we can specify, visualize, and document the components of a software system. And OCL is part of the UML and has been used for object-oriented analysis and design as a de facto constraints specification language in software engineering arena. Our representation is based on a standard model for RBAC proposed by the National Institute of Standards and Technology. We specify this RBAC model with UML including three views: static view, functional view, and dynamic view. We also describe how OCL can specify RBAC constraints that is one of important aspects to constrain what components in RBAC are allowed to do. In addition, we briefly discuss future directions of this work.
11764310
Symmetric drawings of triconnected planar graphs
Symmetry is one of the most important aesthetic criteria in graph drawing because it reveals structure in the graph. To draw graphs symmetrically, we need two steps. The first step is to find appropriate automorphisms. The second step is to draw the graph to display the automorphisms. Our aim in this paper is to construct maximally symmetric straight-line drawings of triconnected planar graphs in linear time. Previously known algorithms run in quadratic time. We show that an algorithm of Fontet can be used to find an embedding in the plane with the maximum number of symmetries, and present a new algorithm for finding a straight line drawing that achieves that maximum. Both algorithms run in linear time.
1395152
A Group-Theoretic Method for Drawing Graphs Symmetrically
Constructing symmetric drawings of graphs is NP-hard. In this paper, we present a new method for drawing graphs symmetrically based on group theory. More formally, we define a n-geometric automorphism group of a graph that can be displayed as symmetries of a drawing of the graph in n dimensions. Then we present an algorithm to find all 2- and 3-geometric automorphism groups of a graph. We implement the algorithm using Magma [11] and the experimental results shows that our approach is very efficient in practice. We also present a drawing algorithm to display a 2- or 3-geometric automorphism group.
40285902
Isoperimetric numbers and bisection widths of double coverings of a complete graph
10549206
Regular graph coverings whose covering transformation groups have the isomorphism extension property
Abstract Enumerative results are presently a major center of interest in topological graph theory, as in the work of Gross and Furst [1], Hofmeister [5,6], Kwak and Lee [9–13] and Mull et al. [15], etc. Kwak and Lee [9] enumerated the isomorphism classes of graph bundles and those of n -fold graph coverings with respect to a group of automorphisms of the base graph which fix a spanning tree. Hofmeister [6] enumerated independently the isomorphism classes of n -fold graph coverings with respect to the trivial automorphism group of the base graph. But the enumeration of isomorphism classes of regular graph coverings has not been answered completely. As its partial answers, Hofmeister enumerated the isomorphism classes of Z 2 -coverings (double coverings) with respect to any group of automorphisms of the base graph, and Sato [14] did the same work for Z p -coverings (regular prime-fold coverings). With respect to the trivial automorphism group of the base graph, Hong and Kwak [8] did the same work for Z 2 ⊕ Z 2 or Z 4 -coverings, and Kwak and Lee [10] did it for Z p , Z p ⊕ Z 1 ( p ≠ q primes) or Z p 2 -coverings. As an expansion of this effort, we obtain in this paper several new algebraic characterizations for isomorphic regular coverings and derive an enumerating formula for the isomorphism classes of A-coverings of a graph G with respect to any group of automorphisms of G which fix a spanning tree, when the covering transformation group A has the isomorphism extension property. By definition, it means that every isomorphism between any two isomorphic subgroups B 1 and B 2 of A can be extended to an automorphism of A. Also, we obtain complete numerical enumeration of the isomorphism classes of Z n -coverings for all n , D n -coverings for odd n (D n is the dihedral group of order 2 n ) or Z p ⊕ Z p -coverings of a graph G for prime p with respect to the trivial automorphism group of G . In addition, we applied our results to a bouquet of circles.