A simple recognition of maximal planar graphs

ArticleinInformation Processing Letters 89(5):223-226 · March 2004with 860 Reads
Abstract
In this paper, we consider the problem of determining whether a given graph is a maximal planar graph or not. We show that a simple linear time algorithm can be designed based on canonical orderings. Our algorithm needs no sophisticated data structure and is significantly easy to implement compared with the existing planarity testing algorithms.

Do you want to read the rest of this article?

Request full-text
Request Full-text Paper PDF
  • ... The concept of a structure for finite or infinite maximal planar graphs with respect to the existence of hamiltonian cycles (or one-way infinite hamiltonian paths) is due to Whitney [11]. It has since inspired some profound work in graph theory; see for example Goddard [4], Jackson and Yu [6], Nagamochi et al. [9], Dean et al. [1], Thomassen [10] or any of several articles in [2] or [8]. The first part of this paper is devoted to investigate the concept of maximal planar graphs. ...
    Article
    In the first part of this paper we investigate several statements con-cerning infinite maximal planar graphs which are equivalent in finite case. In the second one, for a given induced θ-path (a finite induced path whose endvertices are adjacent to a vertex of infinite degree) in a 4-connected VAP-free maximal planar graph containing a vertex of infinite degree, a new θ-path is constructed such that the resulting fan is tight.
  • Article
    The architectural layout design problem, which is concerned with the finding of the best adjacencies between functional spaces among many possible ones under given constraints, can be formulated as a combinatorial optimization problem and can be solved with an Evolutionary Algorithm (EA). We present functional spaces and their adjacencies in form of graphs and propose an EA called EvoArch that works with a graph-encoding scheme. EvoArch encodes topological configuration in the adjacency matrices of the graphs that they represent and its reproduction operators operate on these adjacency matrices. In order to explore the large search space of graph topologies, these reproduction operators are designed to be unbiased so that all nodes in a graph have equal chances of being selected to be swapped or mutated. To evaluate the fitness of a graph, EvoArch makes use of a fitness function that takes into consideration preferences for adjacencies between different functional spaces, budget and other design constraints. By means of different experiments, we show that EvoArch can be a very useful tool for architectural layout design tasks.
  • Article
    Answering a question of Rosenstiehl and Tarjan, we show that every plane graph withn vertices has a Fry embedding (i.e., straight-line embedding) on the 2n–4 byn–2 grid and provide anO(n) space,O(n logn) time algorithm to effect this embedding. The grid size is asymptotically optimal and it had been previously unknown whether one can always find a polynomial sized grid to support such an embedding. On the other hand we show that any setF, which can support a Fry embedding of every planar graph of sizen, has cardinality at leastn+(1–o(1))n which settles a problem of Mohar.
  • Article
    This paper describes an efficient algorithm to determine whether an arbitrary graph G can be embedded in the plane. The algorithm may be viewed as an iterative version of a method originally proposed by L. Auslander and S. V. Parter and correctly formulated by A. J. Goldstein. The algorithm uses depth-first search and has O(V) time and space bounds, where V is the number of vertices in G. An ALGOL implementation of the algorithm successfully tested graphs with as many as 900 vertices in less than 12 seconds.