A Planarity Test via Construction Sequences

Optimal linear-time algorithms for testing the planarity of a graph are well-known for over 35 years. However, these algorithms are quite involved and recent publications still try to give simpler linear-time tests. We give a simple reduction from planarity testing to the problem of computing a certain construction of a 3-connected graph. The approach is different from previous planarity tests; as key concept, we maintain a planar embedding that is 3-connected at each point in time. The algorithm runs in linear time and computes a planar embedding if the input graph is planar and a Kuratowski-subdivision otherwise

Maximum Planar Subgraph on Graphs not Contractive to K5 or K3,3

The maximum planar subgraph problem is well studied. Recently, it has been shown that the maximum planar subgraph problem is NP-complete for cubic graphs. In this paper we prove shortly that the maximum planar subgraph problem remains NP-complete even for graphs without a minor isomorphic to K5 or K3,3 , respectively

Planar subgraphs without low-degree nodes

We study the following problem: given a geometric graph G and an integer k, determine if G has a planar spanning subgraph (with the original embedding and straight-line edges) such that all nodes have degree at least k. If G is a unit disk graph, the problem is trivial to solve for k = 1. We show that even the slightest deviation from the trivial case (e.g., quasi unit disk graphs or k = 1) leads to NP-hard problems.Peer reviewe

Intuitive Visualization and Analysis of Multi-Omics Data and Application to Escherichia coli Carbon Metabolism

Combinations of ‘omics’ investigations (i.e, transcriptomic, proteomic, metabolomic and/or fluxomic) are increasingly applied to get comprehensive understanding of biological systems. Because the latter are organized as complex networks of molecular and functional interactions, the intuitive interpretation of multi-omics datasets is difficult. Here we describe a simple strategy to visualize and analyze multi-omics data. Graphical representations of complex biological networks can be generated using Cytoscape where all molecular and functional components could be explicitly represented using a set of dedicated symbols. This representation can be used i) to compile all biologically-relevant information regarding the network through web link association, and ii) to map the network components with multi-omics data. A Cytoscape plugin was developed to increase the possibilities of both multi-omic data representation and interpretation. This plugin allowed different adjustable colour scales to be applied to the various omics data and performed the automatic extraction and visualization of the most significant changes in the datasets. For illustration purpose, the approach was applied to the central carbon metabolism of Escherichia coli. The obtained network contained 774 components and 1232 interactions, highlighting the complexity of bacterial multi-level regulations. The structured representation of this network represents a valuable resource for systemic studies of E. coli, as illustrated from the application to multi-omics data. Some current issues in network representation are discussed on the basis of this work

Street Network Models and Indicators for Every Urban Area in the World

Cities worldwide exhibit a variety of street network patterns and configurations that shape human mobility, equity, health, and livelihoods. This study models and analyzes the street networks of every urban area in the world, using boundaries derived from the Global Human Settlement Layer. Street network data are acquired and modeled from OpenStreetMap with the open-source OSMnx software. In total, this study models over 160 million OpenStreetMap street network nodes and over 320 million edges across 8,914 urban areas in 178 countries, and attaches elevation and grade data. This article presents the study's reproducible computational workflow, introduces two new open data repositories of ready-to-use global street network models and calculated indicators, and discusses summary findings on street network form worldwide. It makes four contributions. First, it reports the methodological advances of this open-source workflow. Second, it produces an open data repository containing street network models for each urban area. Third, it analyzes these models to produce an open data repository containing street network form indicators for each urban area. No such global urban street network indicator dataset has previously existed. Fourth, it presents a summary analysis of urban street network form, reporting the first such worldwide results in the literature

Planarity and Street Network Representation in Urban Form Analysis

Models of street networks underlie research in urban travel behavior, accessibility, design patterns, and morphology. These models are commonly defined as planar, meaning they can be represented in two dimensions without any underpasses or overpasses. However, real-world urban street networks exist in three-dimensional space and frequently feature grade separation such as bridges and tunnels: planar simplifications can be useful but they also impact the results of real-world street network analysis. This study measures the nonplanarity of drivable and walkable street networks in the centers of 50 cities worldwide, then examines the variation of nonplanarity across a single city. It develops two new indicators - the Spatial Planarity Ratio and the Edge Length Ratio - to measure planarity and describe infrastructure and urbanization. While some street networks are approximately planar, we empirically quantify how planar models can inconsistently but drastically misrepresent intersection density, street lengths, routing, and connectivity

An Improved Algorithm for Finding Maximum Outerplanar Subgraphs

We study the NP-complete Maximum Outerplanar Subgraph problem. The previous best known approximation ratio for this problem is 2/3. We propose a new approximation algorithm which improves the ratio to 7/10

A Linear-Time Algorithm for Finding Induced Planar Subgraphs

In this paper we study the problem of efficiently and effectively extracting induced planar subgraphs. Edwards and Farr proposed an algorithm with O(mn) time complexity to find an induced planar subgraph of at least 3n/(d+1) vertices in a graph of maximum degree d. They also proposed an alternative algorithm with O(mn) time complexity to find an induced planar subgraph graph of at least 3n/(bar{d}+1) vertices, where bar{d} is the average degree of the graph. These two methods appear to be best known when d and bar{d} are small. Unfortunately, they sacrifice accuracy for lower time complexity by using indirect indicators of planarity. A limitation of those approaches is that the algorithms do not implicitly test for planarity, and the additional costs of this test can be significant in large graphs. In contrast, we propose a linear-time algorithm that finds an induced planar subgraph of n-nu vertices in a graph of n vertices, where nu denotes the total number of vertices shared by the detected Kuratowski subdivisions. An added benefit of our approach is that we are able to detect when a graph is planar, and terminate the reduction. The resulting planar subgraphs also do not have any rigid constraints on the maximum degree of the induced subgraph. The experiment results show that our method achieves better performance than current methods on graphs with small skewness