N-vertex Planar Research Articles

On local transformations in plane geometric graphs embedded on small grids

Given two n -vertex plane graphs G 1 = ( V 1 , E 1 ) and G 2 = ( V 2 , E 2 ) with | E 1 | = | E 2 | embedded in the n × n grid, with straight-line segments as edges, we show that with a sequence of O ( n ) point moves (all point moves stay within a 5 n × 5 n grid) and O ( n 2 ) edge moves, we can transform the embedding of G 1 into the embedding of G 2 . In the case of n -vertex trees, we can perform the transformation with O ( n ) point and edge moves with all moves staying in the n × n grid. We prove that this is optimal in the worst case as there exist pairs of trees that require Ω ( n ) point and edge moves. We also study the equivalent problems in the labeled setting.

Colorings Of Plane Graphs With No Rainbow Faces

We prove that each n-vertex plane graph with girth g≥4 admits a vertex coloring with at least ⌈n/2⌉+1 colors with no rainbow face, i.e., a face in which all vertices receive distinct colors. This proves a conjecture of Ramamurthi and West. Moreover, we prove for plane graph with girth g≥5 that there is a vertex coloring with at least $${\left\lceil {\frac{{g - 3}}{{g - 2}}n - \frac{{g - 7}}{{2{\left( {g - 2} \right)}}}} \right\rceil }$$ if g is odd and $${\left\lceil {\frac{{g - 3}}{{g - 2}}n - \frac{{g - 6}}{{2{\left( {g - 2} \right)}}}} \right\rceil }$$ if g is even. The bounds are tight for all pairs of n and g with g≥4 and n≥5g/2−3.

A 1.235 lower bound on the number of points needed to draw all n-vertex planar graphs

We study a problem of lower bounds on straight line drawings of planar graphs. We show that at least 1.235 ċ n points in the plane are required to draw each n-vertex planar graph with edges drawn as straight line segments (for sufficiently large n). This improves the previous best bound of 1.206 ċ n (for sufficiently large n) due to Chrobak and Karloff [Sigact News 20 (4) (1989) 83-86]. Our contribution is twofold: we improve the lower bound itself and we give a significantly simpler and more straightforward proof.

On maximum face-constrained coloring of plane graphs with no short face cycles

We prove that the vertices of each n -vertex plane graph G with minimum face cycle length g, g⩾5 , can be colored using at least g 2 −g−8 g 2 +g−6 n+ 4g+4 g 2 +g−6 colors (for n ⩾( g +3)/2) in such a way that G does not contain a polychromatic face, i.e., a face whose all the vertices have mutually different colors. In particular, if the girth of an n -vertex plane graph is at least five, then there is a coloring using at least ⌈ n /2⌉+1 colors.

Worst-case-optimal algorithms for guarding planar graphs and polyhedral surfaces

We present an optimal Θ(n)-time algorithm for the selection of a subset of the vertices of an n-vertex plane graph G so that each of the faces of G is covered by (i.e., incident with) one or more of the selected vertices. At most ⌊n/2⌋ vertices are selected, matching the worst-case requirement. Analogous results for edge-covers are developed for two different notions of “coverage”. In particular, our linear-time algorithm selects at most n−2 edges to strongly cover G, at most ⌊n/3⌋ diagonals to cover G, and in the case where G has no quadrilateral faces, at most ⌊n/3⌋ edges to cover G. All these bounds are optimal in the worst-case. Most of our results flow from the study of a relaxation of the familiar notion of a 2-coloring of a plane graph which we call a face-respecting 2-coloring that permits monochromatic edges as long as there are no monochromatic faces. Our algorithms apply directly to the location of guards, utilities or illumination sources on the vertices or edges of polyhedral terrains, polyhedral surfaces, or planar subdivisions.

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) × O(n) grid, while maintaining ?(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 C1-continuous curves, represented by a sequence of at most three circular arcs.

A Framework for Drawing Planar Graphs with Curves and Polylines

We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well as complexity measures such as vertex and edge representational complexity and the area of the drawing. In addition to this general framework, we present algorithms that operate within this framework. Specifically, we describe an algorithm for drawing any n-vertex planar graph in an O(n)×O(n) grid using polylines that have at most two bends per edge and asymptotically-optimal worst-case angular resolution. More significantly, we show how to adapt this algorithm to draw any n-vertex planar graph using cubic Bezier curves, with all vertices and control points placed within an O(n)×O(n) integer grid so that the curved edges achieve a curvilinear analogue of good angular resolution. All of our algorithms run in O(n) time.

Dominating Sets in Planar Graphs

Motivated by an application to unstructured multigrid calculations, we consider the problem of asymptotically minimizing the size of dominating sets in triangulated planar graphs. Specifically, we wish to find the smallest ε such that, for n sufficiently large, every n- vertex planar graph contains a dominating set of size at most εn. We prove that 1/4<ε<1/3, and we conjecture that ε=1/4. For triangulated discs we obtain a tight bound of ε=1/3. The upper bound proof yields a linear-time algorithm for finding an (n/3)- size dominating set.

A new approach to solving three combinatorial enumeration problems on planar graphs

The purpose of this paper is to show how the technique of delta-wye graph reduction provides an alternative method for solving three enumerative function evaluation problems on planar graphs. In particular, it is shown how to compute the number of spanning trees and perfect matchings, and how to evaluate energy in the Ising “spin glass” model of statistical mechanics. These alternative algorithms require O( n 2 ) arithmetic operations on an n -vertex planar grapha, and are relatively easy to implement.

The pagewidth of trivalent planar graphs

We prove the following result: there exist trivalent n -vertex planar graphs, any 2-page embedding of which has pagewidth Ω( n ).

On cleaving a planar graph

We first show that the removal of 4√ n / ϵ vertices from an n -vertex planar graph with non-negative vertex weights summing to no more than 1 is sufficient to cleave or recursively separate it into components of weight no more than a given ϵ , thus improving on the 2√6 √ n / ϵ bound shown in [6]. We then derive worst-case bounds on the number of vertices necessary to separate a planar graph of a given radius into components of weight no more than ϵ .

Optimal Parallel 5-Colouring of Planar Graphs

We show that a 5-colouring of the vertices of an n-vertex planar graph may be computed in $O(\log n\log ^ * n)$ time by an exclusive-read exclusive-write parallel RAM with $O({n / {(\log n\log ^ * n)}})$ processors. Our algorithm, while faster than all previously known methods, is at the same time the first parallel 5-colouring algorithm to exhibit an optimal speedup. Optimality is achieved through a method based on the accelerating cascades technique and of independent interest. It should be emphasized that although input to the algorithm is a planar graph, we do not require a planar embedding to be given as part of the input. Other results concern the colouring of graphs of bounded genus and the construction of search structures for triangular planar subdivisions.

A linear-processor algorithm for depth-first search in planar graphs

Abstract We present an n -processor and O(log 2 n ) time parallel RAM algorithm for finding a depth-first-search tree in an n -vertex planar graph. The algorithm is based on a new n -processor algorithm for finding a cyclic separator in a planar graph and Smith's (1986) original parallel depth-first-search algorithm for planar graphs.

On the frequency of 3-connected subgraphs of planar graphs

The concept of dependence of subgraphs of a plane graph is defined, as a measure of how much they overlap. It is shown that if M is a 3-connected plane graph, then the number of copies of M in a plane graph which are dependent on a given copy is bounded above by a constant c (M). The number of copies of M in any n-vertex plane graph is at most nc (m).

Applications of a Planar Separator Theorem

Any n-vertex planar graph has the property that it can be divided into components of roughly equal size by removing only $O(\sqrt n )$ vertices. This separator theorem, in combination with a divide-and-conquer strategy, leads to many new complexity results for planar graph problems. This paper describes some of these results.

A Separator Theorem for Planar Graphs

Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than ${2n / 3}$ vertices, and C contains no more than $2\sqrt 2 \sqrt n $ vertices. We exhibit an algorithm which finds such a partition A, B, C in $O( n )$ time.