Visibility Drawings of Plane 3-Trees with Minimum Area

A visibility drawing of a plane graph G is a drawing of G where each vertex is drawn as a horizontal line segment and each edge is drawn as a vertical line segment such that the line segments use only grid points as their endpoints. The area of a visibility drawing is the area of the smallest rectangle on the grid which encloses the drawing. A minimum-area visibility drawing of a plane graph G is a visibility drawing of G where the area is the minimum among all possible visibility drawings of G. The area minimization for grid visibility representation of planar graphs is NP-hard. However, the problem can be solved for a fixed planar embedding of a hierarchically planar graph in quadratic time. In this paper, we give a polynomial-time algorithm to obtain minimum-area visibility drawings of plane 3-trees.     

Open Rectangle-of-Influence Drawings of Inner Triangulated Plane Graphs

A straight-line drawing of a plane graph is called an open rectangle-of-influence drawing if there is no vertex in the proper inside of the axis-parallel rectangle defined by the two ends of every edge. In an inner triangulated plane graph, every inner face is a triangle although the outer face is not necessarily a triangle. In this paper, we first obtain a sufficient condition for an inner triangulated plane graph G to have an open rectangle-of-influence drawing; the condition is expressed in terms of a labeling of angles of a subgraph of G. We then present an O(n ^1.5/log n)-time algorithm to examine whether G satisfies the condition and, if so, construct an open rectangle-of-influence drawing of G on an (n−1)×(n−1) integer grid, where n is the number of vertices in G.     

Rectangular drawings of plane graphs without designated corners

This paper addresses the problem of finding rectangular drawings of plane graphs, in which each vertex is drawn as a point, each edge is drawn as a horizontal or a vertical line segment, and the contour of each face is drawn as a rectangle. A graph is a 2–3 plane graph if it is a plane graph and each vertex has degree 3 except the vertices on the outer face which have degree 2 or 3. A necessary and sufficient condition for the existence of a rectangular drawing has been known only for the case where exactly four vertices of degree 2 on the outer face are designated as corners in a 2–3 plane graph G. In this paper we establish a necessary and sufficient condition for the existence of a rectangular drawing of G for the general case in which no vertices are designated as corners. We also give a linear-time algorithm to find a rectangular drawing of G if it exists.     

An algorithm for constructing star-shaped drawings of plane graphs

A straight-line planar drawing of a plane graph is called a convex drawing if every facial cycle is drawn as a convex polygon. Convex drawings of graphs is a well-established aesthetic in graph drawing, however not all planar graphs admit a convex drawing. Tutte [W.T. Tutte, Convex representations of graphs, Proc. of London Math. Soc. 10 (3) (1960) 304–320] showed that every triconnected plane graph admits a convex drawing for any given boundary drawn as a convex polygon. Thomassen [C. Thomassen, Plane representations of graphs, in: Progress in Graph Theory, Academic Press, 1984, pp. 43–69] gave a necessary and sufficient condition for a biconnected plane graph with a prescribed convex boundary to have a convex drawing.In this paper, we initiate a new notion of star-shaped drawing of a plane graph as a straight-line planar drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. A star-shaped drawing is a natural extension of a convex drawing, and a new aesthetic criteria for drawing planar graphs in a convex way as much as possible. We give a sufficient condition for a given set A of corners of a plane graph to admit a star-shaped drawing whose concave corners are given by the corners in A, and present a linear time algorithm for constructing such a star-shaped drawing.     

Generalized Thrackle Drawings of Non-bipartite Graphs

A graph drawing is called a generalized thrackle if every pair of edges meets an odd number of times. In a previous paper, we showed that a bipartite graph G can be drawn as a generalized thrackle on an oriented closed surface M if and only if G can be embedded in M. In this paper, we use Lins’ notion of a parity embedding and show that a non-bipartite graph can be drawn as a generalized thrackle on an oriented closed surface M if and only if there is a parity embedding of G in a closed non-orientable surface of Euler characteristic χ(M)−1. As a corollary, we prove a sharp upper bound for the number of edges of a simple generalized thrackle.     

Drawing c-planar biconnected clustered graphs

In a graph, a cluster is a set of vertices, and two clusters are said to be non-intersecting if they are disjoint or one of them is contained in the other. A clustered graph C consists of a graph G and a set of non-intersecting clusters. In this paper, we assume that C has a compound planar drawing and each cluster induces a biconnected subgraph. Then we show that such a clustered graph admits a drawing in the plane such that (i) edges are drawn as straight-line segments with no edge crossing and (ii) the boundary of the biconnected subgraph induced by each cluster is a convex polygon.     

Upward straight-line embeddings of directed graphs into point sets

In this paper we study the problem of computing an upward straight-line embedding of a planar DAG (directed acyclic graph) G into a point set S, i.e. a planar drawing of G such that each vertex is mapped to a point of S, each edge is drawn as a straight-line segment, and all the edges are oriented according to a common direction. In particular, we show that no biconnected DAG admits an upward straight-line embedding into every point set in convex position. We provide a characterization of the family of DAGs that admit an upward straight-line embedding into every convex point set such that the points with the largest and the smallest y-coordinate are consecutive in the convex hull of the point set. We characterize the family of DAGs that contain a Hamiltonian directed path and that admit an upward straight-line embedding into every point set in general position. We also prove that a DAG whose underlying graph is a tree does not always have an upward straight-line embedding into a point set in convex position and we describe how to construct such an embedding for a DAG whose underlying graph is a path. Finally, we give results about the embeddability of some sub-classes of DAGs whose underlying graphs are trees on point set in convex and in general position.     

On embedding a cycle in a plane graph

Consider a planar drawing Γ of a planar graph G such that the vertices are drawn as small circles and the edges are drawn as thin stripes. Consider a non-simple cycle c of G. Is it possible to draw c as a non-intersecting closed curve inside Γ, following the circles that correspond in Γ to the vertices of c and the stripes that connect them? We show that this test can be done in polynomial time and study this problem in the framework of clustered planarity for highly non-connected clustered graphs.     

Convex Drawings of Planar Graphs and the Order Dimension of 3-Polytopes

We define an analogue of Schnyder's tree decompositions for 3-connected planar graphs. Based on this structure we obtain: Let G be a 3-connected planar graph with f faces, then G has a convex drawing with its vertices embedded on the (f–1)×(f–1) grid. Let G be a 3-connected planar graph. The dimension of the incidence order of vertices, edges and bounded faces of G is at most 3.The second result is originally due to Brightwell and Trotter. Here we give a substantially simpler proof.     

Grid Embedding of 4-Connected Plane Graphs

A straight line grid embedding of a plane graph G is a drawing of G such that the vertices are drawn at grid points and the edges are drawn as nonintersecting straight line segments. In this paper we show that if a 4-connected plane graph G has at least four vertices on its external face, then G can be embedded on a grid of size such that and , where n is the number of vertices of G. Such an embedding can be computed in linear time. Received March 30, 1995, and in revised form January 3, 1996.     

Bounds for rectilinear crossing numbers

A rectilinear drawing of a graph is one where each edge is drawn as a straight-line segment, and the rectilinear crossing number of a graph is the minimum number of crossings over all rectilinear drawings. We describe, for every integer k ≥ 4, a class of graphs of crossing number k, but unbounded rectilinear crossing number. This is best possible since the rectilinear crossing number is equal to the crossing number whenever the latter is at most 3. Further, if we consider drawings where each edge is drawn as a polygonal line segment with at most one break point, then the resulting crossing number is at most quadratic in the regular crossing number. © 1993 John Wiley & Sons, Inc.     

Octagonal drawings of plane graphs with prescribed face areas

An orthogonal drawing of a plane graph is called an octagonal drawing if each inner face is drawn as a rectilinear polygon of at most eight (polygonal) vertices and the contour of the outer face is drawn as a rectangle. A slicing graph is obtained from a rectangle by repeatedly slicing it vertically and horizontally. A slicing graph is called a good slicing graph if either the upper subrectangle or the lower one obtained by any horizontal slice will never be vertically sliced, roughly speaking. In this paper we show that every good slicing graph has an octagonal drawing with prescribed face areas, in which the area of each inner face is equal to a prescribed value. Such a drawing has practical applications in VLSI floorplanning. We also give a linear-time algorithm to find such a drawing when a “slicing tree” is given. We furthermore present a sufficient condition for a plane graph to be a good slicing graph.     

Homogeneous factorisations of complete graphs with edge-transitive factors

A factorisation of a complete graph K _n is a partition of its edges with each part corresponding to a spanning subgraph (not necessarily connected), called a factor. A factorisation is called homogeneous if there are subgroups M<G≤S _n such that M is vertex-transitive and fixes each factor setwise, and G permutes the factors transitively. We classify the homogeneous factorisations of K _n for which there are such subgroups G,M with M transitive on the edges of a factor as well as the vertices. We give infinitely many new examples. This paper forms part of an Australian Research Council Discovery Grant project, and was a major part of the PhD project of the second author.     

Monotone drawings of planar graphs

Let G be a graph drawn in the plane so that its edges are represented by x‐monotone curves, any pair of which cross an even number of times. We show that G can be redrawn in such a way that the x‐coordinates of the vertices remain unchanged and the edges become non‐crossing straight‐line segments. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 39–47, 2004     

Long cycles and spanning subgraphs of locally maximal 1-planar graphs

A graph is 1-planar if it has a drawing in the plane such that each edge is crossed at most once by another edge. Moreover, if this drawing has the additional property that for each crossing of two edges the end vertices of these edges induce a complete subgraph, then the graph is locally maximal 1-planar. For a 3-connected locally maximal 1-planar graph G, we show the existence of a spanning 3-connected planar subgraph and prove that G is Hamiltonian if G has at most three 3-vertex-cuts, and that G is traceable if G has at most four 3-vertex-cuts. Moreover, infinitely many nontraceable 5-connected 1-planar graphs are presented.     

Biplanar crossing numbers. II. Comparing crossing numbers and biplanar crossing numbers using the probabilistic method

The biplanar crossing number cr₂(G) of a graph G is min{cr(G₁) + cr(G₂)}, where cr is the planar crossing number. We show that cr₂(G) ≤ (3/8)cr(G). Using this result recursively, we bound the thickness by Θ(G) ‐ 2 ≤ Kcr₂(G)^0.4057 log₂n with some constant K. A partition realizing this bound for the thickness can be obtained by a polynomial time randomized algorithm. We show that for any size exceeding a certain threshold, there exists a graph G of this size, which simultaneously has the following properties: cr(G) is roughly as large as it can be for any graph of that size, and cr₂(G) is as small as it can be for any graph of that size. The existence is shown using the probabilistic method. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Bold graph drawings

When a graph is drawn in a classical manner, its vertices are shown as small disks and its edges with a positive width; zero-width edges and zero-size vertices exist only in theory. Let r denote the radius of the disks that show vertices and w the width of edges. We give a list of conditions that make such a drawing good and that apply to not necessarily planar graphs. We show that if r<w, a vertex must have constant degree for a drawing to satisfy the conditions, and if r?w, a vertex can have any degree. We also give an algorithm that, for a given drawing and values for r and w, determines whether the bold drawing satisfies the conditions. Furthermore, we show how to maximize r and/or w without violating the conditions in polynomial time.     

Grid Drawings of 4-Connected Plane Graphs

A grid drawing of a plane graph G is a drawing of G on the plane so that all vertices of G are put on plane grid points and all edges are drawn as straight line segments between their endpoints without any edge-intersection. In this paper we give a very simple algorithm to find a grid drawing of any given 4-connected plane graph G with four or more vertices on the outer face. The algorithm takes time O(n) and yields a drawing in a rectangular grid of width \lceil n/2 \rceil - 1 and height \lfloor n/2\rfloor if G has n vertices. The algorithm is best possible in the sense that there are an infinite number of 4-connected plane graphs, any grid drawings of which need rectangular grids of width \lceil n/2 \rceil - 1 and height \lfloor n/2\rfloor . Received October 13, 1999, and in revised form July 18, 2000. Online publication February 26, 2001.     

Infinitely many pairs of cospectral integral regular graphs

A graph G is called integral if all the eigenvalues of the adjacency matrix A(G) of G are integers. In this paper, the graphs G ₄(a, b) and G ₅(a, b) with 2a+6b vertices are defined. We give their characteristic polynomials from matrix theory and prove that the (n+2)-regular graphs G ₄(n, n+2) and G ₅(n, n+2) are a pair of non-isomorphic connected cospectral integral regular graphs for any positive integer n.