Planarity Testing Problem Research Articles

A fast detection method to solve mechanical interference problem in automatic transmission

In the process of configuration synthesis of automatic transmission (AT), it is difficult to eliminate configurations with mechanical interference because the methods used to detect this problem are often not fast and accurate enough. In this paper, we propose a method to fast eliminate the mechanical interference results of ATs with planetary gear train (PGT). A unified digital model is proposed to represent ATs. The digital model can be automatically converted into matrix representation of the detection object. Then, the problem of mechanical interference detection can be transformed into planarity testing problem by matrix representation. Therefore, combining Hopcroft-Tarjan planarity testing algorithm and matrix representation enables fast detection of mechanical interference in AT. Taking some classical AT as examples, the implementation process is illustrated in detail and the accuracy of the results is verified. After testing, the average time taken for a single detection is 0.7913 ms, so it can be used to detect the results of large quantity configuration synthesis. Furthermore, this method can also be integrated in the AT mechanism intelligent design software to assist in the design.

Simultaneous FPQ-ordering and hybrid planarity testing

We introduce and study a constrained planarity testing problem, called 1-Fixed Constrained Planarity, and prove that this problem can be solved in quadratic time for biconnected graphs. Our solution is based on a novel definition of fixedness that makes it possible to simplify and extend known techniques about Simultaneous PQ-Ordering. We exploit this result to study different versions of the hybrid planarity testing problem. Namely, we show polynomial-time solutions for a variant of NodeTrix Planarity with fixed sides, for PolyLink Planarity, and for Clique Planarity with fixed sides.

NodeTrix Planarity Testing with Small Clusters

We study the NodeTrix planarity testing problem for flat clustered graphs when the maximum size of each cluster is bounded by a constant $k$. We consider both the case when the sides of the matrices to which the edges are incident are fixed and the case when they can be chosen arbitrarily. We show that NodeTrix planarity testing with fixed sides can be solved in $O(k^{3k+\frac{3}{2}} \cdot n)$ time for every flat clustered graph that can be reduced to a partial 2-tree by collapsing its clusters into single vertices. In the general case, NodeTrix planarity testing with fixed sides can be solved in $O(n)$ time for $k = 2$, but it is NP-complete for any $k > 2$. NodeTrix planarity testing remains NP-complete also in the free sides model when $k > 4$.

Strip Planarity Testing for Embedded Planar Graphs

In this paper we introduce and study the strip planarity testing problem, which takes as an input a planar graph G(V, E) and a function $$\gamma :V \rightarrow \{1,2,\dots ,k\}$$ and asks whether a planar drawing of G exists such that each edge is represented by a curve that is monotone in the y-direction and, for any $$u,v\in V$$ with $$\gamma (u)<\gamma (v)$$ , it holds that $$y(u)<y(v)$$ . The problem has strong relationships with some of the most deeply studied variants of the planarity testing problem, such as clustered planarity, upward planarity, and level planarity. Most notably, we provide a polynomial-time reduction from strip planarity testing to clustered planarity. We show that the strip planarity testing problem is polynomial-time solvable if G has a prescribed combinatorial embedding.

On the Upward Planarity of Mixed Plane Graphs

A mixed plane graph is a plane graph whose edge set is partitioned into a set of directed edges and a set of undirected edges. An orientation of a mixed plane graph G is an assignment of directions to the undirected edges of G resulting in a directed plane graph $\vec G$ . In this paper, we study the computational complexity of testing whether a given mixed plane graph G is upward planar, i.e., whether it admits an orientation resulting in a directed plane graph G such that G admits a planar drawing in which each edge is represented by a curve monotonically increasing in the y-direction according to its orientation. Our contribution is threefold. First, we show that the upward planarity testing problem is solvable in cubic time for mixed outerplane graphs. Second, we show that the problem of testing the upward planarity of mixed plane graphs reduces in quadratic time to the problem of testing the upward planarity of mixed plane triangulations. Third, we exhibit linear-time testing algorithms for two classes of mixed plane triangulations, namely mixed plane 3-trees and mixed plane triangulations in which the undirected edges induce a forest.

Switch-Regular Upward Planarity Testing of Directed Trees

Upward planar drawings of digraphs are crossing free drawings where all edges flow in the upward direction. The problem of deciding whether a digraph admits an upward planar drawing is called the upward planarity testing problem, and it has been widely studied in the literature. In this paper we investigate a new upward planarity testing problem, that is, deciding whether a digraph admits an upward planar drawing having some special topological properties: such a drawing is called switchregular. Switch-regular upward planar drawings have practical algorithmic impacts in several graph drawing applications. We provide characterizations for the class of directed trees that admit a switch-regular upward planar drawing. Based on these characterizations we describe an optimal linear-time testing and embedding algorithm.

Overlapping Cluster Planarity

Cluster planarity is currently recognized as one of the most interesting problem in graph drawing. This paper investigates a new direction in this area by addressing the following question: let G be a graph along with a hierarchy of vertex clusters, where clusters can partially intersect. Does G admit a drawing where each cluster is inside a simple closed region, no two edges intersect, and no edge intersects a region twice? We investigate the interplay between this problem and the classical cluster planarity testing problem where clusters are not allowed to partially intersect. Characterizations, models, and algorithms are discussed.

Level Planar Embedding in Linear Time

In a level directed acyclic graph G = (V;E) the vertex set V is partitioned into k ≤ |V | levels V1; V2... Vk such that for each edge (u, v) ∈ E with u ∈ Vi and v ∈; Vj we have i < j. The level planarity testing problem is to decide if G can be drawn in the plane such that for each level Vi, all v ∈ Vi are drawn on the line li = {(x, k - i) | x ∈ ℝ}, the edges are drawn monotonically with respect to the vertical direction, and no edges intersect except at their end vertices. In order to draw a level planar graph without edge crossings, a level planar embedding of the level graph has to be computed. Level planar embeddings are characterized by linear orderings of the vertices in each Vi (1 ≤ i ≤ k). We present an O(|V |) time algorithm for embedding level planar graphs. This approach is based on a level planarity test by Junger, Leipert, and Mutzel [6].

Incremental Convex Planarity Testing

An important class of planar straight-line drawings of graphs are convex drawings, in which all the faces are drawn as convex polygons. A planar graph is said to be convex planar if it admits a convex drawing. We give a new combinatorial characterization of convex planar graphs based on the decomposition of a biconnected graph into its triconnected components. We then consider the problem of testing convex planarity in an incremental environment, where a biconnected planar graph is subject to on-line insertions of vertices and edges. We present a data structure for the on-line incremental convex planarity testing problem with the following performance, where n denotes the current number of vertices of the graph: (strictly) convex planarity testing takes O(1) worst-case time, insertion of vertices takes O(log n) worst-case time, insertion of edges takes O(log n) amortized time, and the space requirement of the data structure is O(n).

A new planarity test

Given an undirected graph, the planarity testing problem is to determine whether the graph can be drawn in a plane without any crossing edges. Linear time planarity testing algorithms have previously been designed by Hopcroft and Tarjan, and by Booth and Lueker. However, their approaches are quite involved. Several other approaches have also been developed for simplifying the planarity test. In this paper, we developed a very simple linear time testing algorithm based only on a depth-first search tree. When the given graph is not planar, our algorithm immediately produces explicit Kuratowski's subgraphs. A new data structure, PC-trees, is introduced, which can be viewed as abstract subembeddings of actual planar embeddings. A graph-reduction technique is adopted so that the embeddings for the planar biconnected components constructed at each iteration never have to be changed. The recognition and embedding are actually done simultaneously in our algorithm (Booth and Lueker, 1976). The implementation of our algorithm is quite straightforward.