The DFS-heuristic for orthogonal graph drawing

In this paper, we present a new heuristic for orthogonal graph drawings, which creates drawings by performing a depth-first search and placing the nodes in the order they are encountered. This DFS-heuristic works for graphs with arbitrarily high degrees, and particularly well for graphs with maximum degree 3. It yields drawings with at most one bend per edge, and a total number of m−n+1 bends for a graph with n nodes and m edges; this improves significantly on the best previous bound of m−2 bends.     

On embedding an outer-planar graph in a point set

Given an n-vertex outer-planar graph G and a set P of n points in the plane, we present an O(nlog³n) time and O(n) space algorithm to compute a straight-line embedding of G in P, improving upon the algorithm in [8,12] that requires O(n²) time. Our algorithm is near-optimal as there is an Ω(nlogn) lower bound for the problem [4]. We present a simpler O(nd) time and O(n) space algorithm to compute a straight-line embedding of G in P where lognd2n is the length of the longest vertex disjoint path in the dual of G. Therefore, the time complexity of the simpler algorithm varies between O(nlogn) and O(n²) depending on the value of d. More efficient algorithms are presented for certain restricted cases. If the dual of G is a path, then an optimal Θ(nlogn) time algorithm is presented. If the given point set is in convex position then we show that O(n) time suffices.     

Drawings of planar graphs with few slopes and segments

We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on n vertices has a plane drawing with at most segments and at most 2n slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of non-planar graphs with few slopes are also considered.     

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.     

Area-efficient planar straight-line drawings of outerplanar graphs

It is important to minimize the area of a drawing of a graph, so that the drawing can fit in a small drawing-space. It is well-known that a planar graph with n vertices admits a planar straight-line grid drawing with O(n²) area [H. de Fraysseix, J. Pach, R. Pollack, How to draw a planar graph on a grid, Combinatorica 10(1) (1990) 41-51; W. Schnyder, Embedding planar graphs on the grid, in: Proceedings of the First ACM-SIAM Symposium on Discrete Algorithms, 1990, pp. 138-148]. Unfortunately, there is a matching lower-bound of Ω(n²) on the area-requirements of the planar straight-line grid drawings of certain planar graphs. Hence, it is important to investigate important categories of planar graphs to determine if they admit planar straight-line grid drawings with o(n²) area.In this paper, we investigate an important category of planar graphs, namely, outerplanar graphs. We show that an outerplanar graph G with degree d admits a planar straight-line grid drawing with area O(dn^1.48) in O(n) time. This implies that if d=o(n^0.52), then G can be drawn in this manner in o(n²) area.     

Graph drawings with few slopes

The slope-number of a graph G is the minimum number of distinct edge slopes in a straight-line drawing of G in the plane. We prove that for Δ5 and all large n, there is a Δ-regular n-vertex graph with slope-number at least . This is the best known lower bound on the slope-number of a graph with bounded degree. We prove upper and lower bounds on the slope-number of complete bipartite graphs. We prove a general upper bound on the slope-number of an arbitrary graph in terms of its bandwidth. It follows that the slope-number of interval graphs, cocomparability graphs, and AT-free graphs is at most a function of the maximum degree. We prove that graphs of bounded degree and bounded treewidth have slope-number at most . Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.     

Two-variable linear programming in parallel

Two-variable linear programming is a fundamental problem in computational geometry. Sequentially, this problem was solved optimally in linear time by Megiddo and Dyer using the elegant prune-and-search technique. In parallel, the previously best known deterministic algorithm on the EREW PRAM for this problem takes O(lognloglogn) time and O(n) work. In this paper, we present a faster parallel deterministic two-variable linear programming algorithm, which takes O(lognlog^*n) time and O(n) work on the EREW PRAM. Our algorithm is based on an interesting parallel prune-and-search technique, and makes use of new geometric observations which can be viewed as generalizations of those used by Megiddo and Dyer's sequential algorithms. Our parallel prune-and-search technique also leads to efficient EREW PRAM algorithm for the weighted selection problem, and is likely to be useful in solving other problems.     

An edge version of the matrix-tree theorem and the wiener index

Let T be a tree on n vertices. The Laplacian matrix is L(T)=D(T)-A(T), where D(T) is the diagonal matrix of vertex degrees and A(T) is the adjacency matrix. A special case of the Matrix-Tree Theorem is that the adjugate of L(T) is the n-by-n matrix of l's. The (n-l)-square "edge version" of L(T)is K(T). The main result is a graph-theoretic interpretation of the entries of the adjugate of K(T). As an application, it is shown that the Wiener Index from chemistry is the trace of this adjugate.     

Embedding complete trees into the hypercube

We consider embeddings of the complete t-ary trees of depth k (denotation T^k,t) as subgraphs into the hypercube of minimum dimension n. This n, denoted by dim(T^k,t), is known if max{k,t}2. First, we study the next open cases t=3 and k=3. We improve the known upper bound dim(T^k,3)2k+1 up to lim_k→∞dim(T^k,3)/k5/3 and show lim_t→∞dim(T^3,t)/t=227/120. As a co-result, we present an exact formula for the dimension of arbitrary trees of depth 2, as a function of their vertex degrees. These results and new techniques provide an improvement of the known upper bound for dim(T^k,t) for arbitrary k and t.     

An elementary algorithm for reporting intersections of red/blue curve segments

Let E_r and E_b be two sets of x-monotone and non-intersecting curve segments, E=E_rE_b and |E|=n. We give a new sweep-line algorithm that reports the k intersecting pairs of segments of E. Our algorithm uses only three simple predicates that allow to decide if two segments intersect, if a point is left or right to another point, and if a point is above, below or on a segment. These three predicates seem to be the simplest predicates that lead to subquadratic algorithms. Our algorithm is almost optimal in this restricted model of computation. Its time complexity is O(nlogn+kloglogn) and it requires O(n) space.     

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.     

A linear time algorithm for edge coloring of binomial trees

We consider the problem of efficient coloring of the edges of a so-called binomial tree T, i.e. acyclic graph containing two kinds of edges: those which must have a single color and those which are to be colored with L consecutive colors, where L is an arbitrary integer greater than 1. We give an O(n) time algorithm for optimal coloring of such a tree, where n is the number of vertices of T. Also, we give simple bounds on the chromatic index of T and a division of all binomial trees into two classes depending on their chromaticity.     

Moore-penrose inverse of the incidence matrix of a tree

Let T be a tree with n vertices, where each edge is given an orientation, and let Q be its vertex-edge incidence matrix. It is shown that the Moore-Penrose inverse of Q is the (n-1)× n matrix M obtained as follows. The rows and the columns of M are indexed by the edges and the vertices of T respectively. If e,ν are an edge and a vertex of T respectively, then the (e,ν)-entry of M is, upto a sign, the number of vertices in the connected component of T\e which does not contain ν. Furthermore, the sign of the entry is positive or negative, depending on whether e is oriented away from or towards ν. This result is then used to obtain an expression for the Moore-Penrose inverse of the incidence matrix of an arbitrary directed graph. A recent result due to Moon is also derived as a consequence.     

A reversal index for tournament matrices

Given a tournament matrix T, its reversal indexi_R(T), is the minimum k such that the reversal of the orientation of k arcs in the directed graph associated with T results in a reducible matrix. We give a formula for i_R(T) in terms of the score vector of T which generalizes a simple criterion for a tournament matrix to be irreducible. We show that i_R(T)≤[(n-1)/2] for any tournament matrix T of order n, with equality holding if and only if T is regular or almost regular, according as n is odd or even. We construct, for each k between 1 and [(n-1)/2], a tournament matrix of order n whose reversal index is k. Finally, we suggest a few problems.     

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.     

Linear operators preserving unitarily invariant norms of matrices

Let F_{m × n} be the set of all m × n matrices over the field F = C or R Denote by U_n(F) the group of all n × n unitary or orthogonal matrices according as F = C or F-R. A norm N() on Fm ×n, is unitarily invariant if N(UAV) = N(A): for all A ∈ F _m×n U ∈ U_m(F). and V ∈ U_n(F). We characterize those linear operators TF_{m × n} → F_{m × n}which satisfy N (T(A)) = N(A)for all A ∈ F_{m × n}

for a given unitarily invariant norm N(). It is shown that the problem is equivalent to characterizing those operators which preserve certain subsets in F_{m × n} To develop the theory we prove some results concerning unitary operators on F_{m × n} which are of independent interest.     

Drawing Trees with Perfect Angular Resolution and Polynomial Area

We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node $v$ equal to $2\pi /d(v)$ . We show:

1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area.
2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution.
3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area.

Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.     

A faster 2-approximation algorithm for the minmax p-traveling salesmen problem on a tree

Given an edge-weighted tree T and an integer p1, the minmax p-traveling salesmen problem on a tree T asks to find p tours such that the union of the p tours covers all the vertices. The objective is to minimize the maximum of length of the p tours. It is known that the problem is NP-hard and has a (2−2/(p+1))-approximation algorithm which runs in O(p^p−1n^p−1) time for a tree with n vertices. In this paper, we consider an extension of the problem in which the set of vertices to be covered now can be chosen as a subset S of vertices and weights to process vertices in S are also introduced in the tour length. For the problem, we give an approximation algorithm that has the same performance guarantee, but runs in O((p−1)!·n) time.     

Drawing planar graphs with circular arcs

In this paper we address the problem of drawing planar graphs with circular arcs while maintaining good angular resolution and small drawing area. We present a lower bound on the area of drawings in which edges are drawn using exactly one circular arc. We also give an algorithm for drawing n -vertex planar graphs such that the edges are sequences of two continuous circular arcs. The algorithm runs in O(n) time and embeds the graph on the O(n)\times O(n) grid, while maintaining Θ(1/d(v)) angular resolution, where d(v) is the degree of vertex v . Since in this case we use circular arcs of infinite radius, this is also the first algorithm that simultaneously achieves good angular resolution, small area, and at most one bend per edge using straight-line segments. Finally, we show how to create drawings in which edges are smooth C ¹ -continuous curves, represented by a sequence of at most three circular arcs. Received September 30, 1999, and in revised form March 27, 2000. Online publication October 26, 2000.