An algorithm for planarity testing by adjacency matrix
15 Jul 2012-Vol. 4, pp 1412-1417
TL;DR: A new method to determine the planarity of a graph by adjacency matrix is proposed, which is very easy to implement and given the degree conditions for the planularity of 6-vertex undirected simple graphs.
Abstract: It is of great use to determine whether a graph is planar in both information technology and engineering areas. Although there are some known algorithms, they are quite difficult to understand and to implement. This paper proposes a new method to determine the planarity of a graph by adjacency matrix, which is very easy to implement. Especially, we give the degree conditions for the planarity of 6-vertex undirected simple graphs.
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01 Feb 2018
TL;DR: Inspired by the classic Page rank algorithm used in web searching, a novel method, named Parking rank, to sort the parking lots based on the public information totally, such as price, location, total spaces, etc, is proposed.
Abstract: Nowadays, it's difficult to find available parking spaces in big cities within China due to the rapid growth of the vehicles. Parking guidance system(PGS) would play an important role to reduce the time spent on looking for parking spaces. An intuitional idea is to collect the real-time occupancy information of city-wide parking lots and guide the vehicles to the proper one on demands. But the idea is hardly implemented practically because of the huge financial and time costs on linking all city-wide parking lots in early deployment stage. Unlike the expensive real-time data, the public information of parking lots is easily obtained with little costs. What if we dig these free public data and use them in parking recommendation? Inspired by the classic Page rank algorithm used in web searching, this paper propose a novel method, named Parking rank, to sort the parking lots based on the public information totally, such as price, location, total spaces, etc. And a driving-cost sensitive recommendation method is presented in this paper. The simulation shows, when working with the Parking rank algorithm, the recommendation algorithm can help vehicles find the proper parking lots efficiently and reasonably, even in the urban district lack of any real-time information. In fact, Parking rank algorithm can be used as pre-processing of static ranking of city-wide parking lots, and can work with other recommendation algorithm related to real-time occupancy information, or work alone if there is only the public information about parking lots.
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982 citations
"An algorithm for planarity testing ..." refers background in this paper
...Kuratowski [1] proved the conclusion that a graph is planar if and only if, after contracting edges, it does not contain a sub-graph K5 or K3,3' Now, this conclusion is known as Kuratowski's theorem....
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TL;DR: This paper improves upon the conference proceedings formulation and upon the Shih-Hsu PC-tree, both of which perform comprehensive tests of planarity conditions embedding the edges from a vertex to its descendants in a ‘batch’ vertex addition operation.
Abstract: We present new O(n)-time methods for planar embedding and Kuratowski subgraph isolation that were inspired by the Booth-Lueker PQ-tree implementation of the Lempel-Even-Cederbaum vertex addition method. In this paper, we improve upon our conference proceedings formulation and upon the Shih-Hsu PC-tree, both of which perform comprehensive tests of planarity conditions embedding the edges from a vertex to its descendants in a ‘batch’ vertex addition operation. These tests are simpler than but analogous to the templating scheme of the PQ-tree. Instead, we take the edge to be the fundamental unit of addition to the partial embedding while preserving planarity. This eliminates the batch planarity condition testing in favor of a few localized decisions of a path traversal process, and it exploits the fact that subgraphs can become biconnected by adding a single edge. Our method is presented using only graph constructs, but our definition of external activity, path traversal process and theoretical analysis of correctness can be applied to optimize the PC-tree
175 citations
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TL;DR: A very simple linear time testing algorithm based only on a depth-first search tree that produces explicit Kuratowski's subgraphs when the given graph is not planar and a graph-reduction technique is adopted so that the embeddings for the planar biconnected components constructed at each iteration never have to be changed.
106 citations
"An algorithm for planarity testing ..." refers background in this paper
...The algorithm was then improved in [4] and [7], and collected in the book [2]....
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Proceedings Article•
01 Jan 1999TL;DR: In this paper, the authors proposed a linear time planar graph embedding algorithm which avoids the need to first st-number the vertices and avoids the use of P-nodes by not connecting pieces together until they become biconnected.
Abstract: A graph is planar if it can be drawn on the plane with no crossing edges. There are several linear time planar embedding algorithms but ah are considered by many to be quite complicated. This paper presents a new method for performing linear time planar graph embedding which avoids some of the complexities of previous approaches (including the need to first st-number the vertices). Our new algorithm easily permits the extraction of a planar obstruction (a subgraph homeomorphic to Ks,s or Ks) in O(n) time if the graph is not planar. Our algorithm is similar to the algorithm of Booth and Lueker which uses a data structure called a PQ-tree. The Pnodes in a P&-tree represent parts of the partially embedded graph that can be permuted, and the Q-nodes represent parts that can be flipped. We avoid the use of P-nodes by not connecting pieces together until they become biconnected. We avoid Q nodes by using a data structure which allows biconnected components to be fiipped in O(1) time.
55 citations
TL;DR: Lapoire solved the conjecture that the treewidth of a planar graph and the treEWidth of its geometric dual differ by at most one using algebraic techniques, and gives a much shorter proof of this result.
37 citations
"An algorithm for planarity testing ..." refers background in this paper
...reader is referred to [5], [6], [11], [13-16]....
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