Linear-Time Algorithms for Problems on Planar Graphs of Fixed Disk Dimension
نویسندگان
چکیده
The disk dimension of a planar graph G is the least number k for which G embeds in the plane minus k open disks, with every vertex on the boundary of some disk. Useful properties of graphs with a given disk dimension are derived, leading to an efficient algorithm to obtain an outerplanar subgraph of a graph of disk dimension k by removing at most 2k−2 vertices. This reduction is used to obtain linear-time exact and approximation algorithms for problems on graphs of fixed disk dimension. In particular, a linear-time 3-approximation algorithm is presented for the pathwidth problem on graphs of fixed disk dimension. This approximation ratio was previously known only for outerplanar graphs (graphs of disk dimension
منابع مشابه
Faces in the Crowd: On the Algorithmic Utility of Disks and Covers
We study a pair of NP-hard problems aimed at finding small sets of faces in planar graphs. In the disk dimension problem, we are given a planar graph G, and seek the least number k for which G embeds in the plane minus k open disks, with every vertex on the boundary of some disk. Useful properties of graphs with bounded disk dimension are derived. We show how to obtain an outerplanar subgraph o...
متن کاملPolynomial Kernels for Hard Problems on Disk Graphs
Kernelization is a powerful tool to obtain fixed-parameter tractable algorithms. Recent breakthroughs show that many graph problems admit small polynomial kernels when restricted to sparse graph classes such as planar graphs, bounded-genus graphs or H-minor-free graphs. We consider the intersection graphs of (unit) disks in the plane, which can be arbitrarily dense but do exhibit some geometric...
متن کاملApproximation Algorithms for Unit Disk Graphs
Mobile ad hoc networks are frequently modeled by unit disk graphs. We consider several classical graph theoretic problems on unit disk graphs (Maximum Independent Set, Minimum Vertex Cover, and Minimum (Connected) Dominating Set), which are relevant to such networks. We propose two new notions for unit disk graphs: thickness and density. The thickness of a graph is the number of disk centers in...
متن کاملBidimensionality and geometric graphs
Bidimensionality theory was introduced by Demaine et al. [JACM 2005 ] as a framework to obtain algorithmic results for hard problems on minor closed graph classes. The theory has been sucessfully applied to yield subexponential time parameterized algorithms, EPTASs and linear kernels for many problems on families of graphs excluding a fixed graph H as a minor. In this paper we use several of th...
متن کاملLinear-Time Algorithms for Parametric Minimum Spanning Tree Problems on Planar Graphs
A linear-time algorithm for the minimum-ratio spanning tree problem on planar graphs is presented. The algorithm is based on a new planar minimum spanning tree algorithm. The approach extends to other parametric minimum spanning tree problems on planar graphs and to other families of graphs having small separators. Disciplines Theory and Algorithms This article is available at Iowa State Univer...
متن کامل