Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with on-line algorithms.

The Steiner tree problem in graphs requires to find a minimum size connected subgraph containing a given subset of nodes (given points). In this paper we consider this problem in three classes of graphs: where the maximum path distance is 2, where given points form a dominating set and where the given points form a vertex cover. As all these problems are MAX-SNP hard, the issue is what approximation can be obtained in polynomial time. In the first case we obtain an approximation ratio (of the achieved size over the minimal one) 5 4 + 3 80 = 1:2875, in the second case we achieve 4/3, and in the last case we achieve 8=7 \Gamma 1=160. 1 Introduction One of the strongest results in matroid theory is the polynomial time solution of the parity problem in linear matroids. Briefly stated, we have a collection of 2 \Theta n matrices, and our goal is to find a maximum size sub-collection such that they can be stacked into a single matrix of maximum rank. This problem was solved by Lov'asz [4]...

We provide the first nontrivial approximation algorithm for MAXIMUM WEIGHT PLANAR SUBGRAPH, the NP-Hard problem of finding a heaviest planar subgraph in an edge-weighted graph G. This problem has applications in circuit layout, facility layout, and graph drawing. No previous algorithm for MAXIMUM WEIGHT PLANAR SUBGRAPH had performance ratio exceeding 1=3, which is obtained by any algorithm that produces a maximum weight spanning tree in G. Based on the Berman-Ramaiyer Steiner tree algorithm, the new algorithm has performance ratio at least 1/3 + 1/72. We also show that if G is complete and its edge weights satisfy the triangle inequality, then the performance ratio is at least 3/8. Furthermore, we derive the first nontrivial performance ratio (7/12 instead of 1/2) for the NP-Hard MAXIMUM WEIGHT OUTERPLANAR SUBGRAPH problem.

Abstract. We construct an optimal linear-time algorithm for the maximal planar subgraph problem: given a graph G, find a planar subgraph G ′ of G such that adding to G ′ an extra edge of G results in a non-planar graph. Our solution is based on a fast data structure for incremental planarity testing of triconnected graphs and a dynamic graph search procedure. Our algorithm can be transformed into a new optimal planarity testing algorithm. Key words. Planar graphs, planarity testing, incremental algorithms, graph planarization, data structures, triconnectivity. AMS subject classifications. 05C10, 05C85, 68R10, 68Q25, 68W40 1. Introduction. Agraphisplanar

. The maximal linear forest problem is to find, given a graph G = (V; E), a maximal subset of V that induces a linear forest. Three parallel algorithms for this problem are presented. The first one is randomized and runs in O(log n) expected time using n 2 processors on a CRCW PRAM. The second one is deterministic and runs in O(log 2 n) time using n 4 processors on an EREW PRAM. The last one is deterministic and runs in O(log 5 n) time using n 3 processors on an EREW PRAM. The results put the problem in the class NC. Keywords: parallel algorithms, randomized parallel algorithms, graph algorithms, linear forests, maximal matchings, maximal independent sets. 1 Introduction Since Karp and Wigderson showed that the maximal independent set (MIS) problem is in the class NC [11], much work has been devoted to the study of parallel complexity of maximality problems. A typical maximality problem on graphs is to find either a maximal vertex-induced subgraph (MVIS) or a maximal edge-...

The maximum planar subgraph, maximum outerplanar subgraph, the thickness and outerthickness of a graph are all NP-complete optimization problems. Apptopinv is a program that contains different heuristic algorithms for these four problems: algorithms based on Hopcroft-Tarjan planarity testing algorithm, the spanning-tree heuristic and various algorithms based on the cactus-tree heuristic. Apptopinv contains also a simulated annealing algorithm that can be used to improve the solutions obtained from other heuristics. Most of the heuristics have also a greedy version. We have implemented graph generators for complete graphs, complete kpartite graphs, complete hypercubes, random graphs, random maximum planar and outerplanar graphs and random regular graphs. Apptopinv supports three different graph file formats. Apptopinv is written in C++ programming language for Linux-platform and GCC 2.95.3 compiler. To compile the program, a commercial LEDA algorithm

. We consider the MAXIMUM PLANAR SUBGRAPH problem - given a graph G, find a largest planar subgraph of G. This problem has applications in circuit layout, facility layout, and graph drawing. We improve to 4/9 the best known approximation ratio for the MAXIMUM PLANAR SUBGRAPH problem. We also consider a generalization of the previous problem, the MAXIMUM GENUS D SUBGRAPH problem - given a connected graph G, find a maximum subgraph of G of genus at most D. For the latter problem, we present a simple algorithm whose approximation ratio is 1/4. 1 Introduction The MAXIMUM PLANAR SUBGRAPH problem is: given a graph G, find a subgraph of G of maximum size, where size is the number of edges. This problem has applications in circuit layout, facility layout, and graph drawing [F92, TDB88]. MAXIMUM PLANAR SUBGRAPH is known to be NP-Complete [LG77]. Therefore we are looking for polynomial-time approximation algorithms. For a graph G, let Opt(G) be the maximum size of a planar subgraph of G. Given ...

The Steiner tree problem in unweighted graphs requires to find a minimum size connected subgraph containing a given subset of nodes (terminals). In this paper we investigate applications of the linear matroid parity algorithm to the Steiner tree problem for two classes of graphs: where the terminals form a vertex cover and where terminals form a dominating set. As all these problems are MAX-SNP-hard, the issue is what approximation can be obtained in polynomial time. The previously best approximation ratio for the first class of graphs (also known as unweighted quasi-bipartite graphs) is ≈ 1.217 (Gröpl et al. [4]) is reduced in this paper to 8/7 − 1/160 ≈ 1.137. For the case of graphs where terminals form a dominating set, an approximation ratio of 4/3 is achieved.

We prove that both the NP-completeness of the nonplanar deletion decision problem and the Max SNP-hardness of the nonplanar deletion problem remain true even for cubic graphs. We prove that the class of graphs with splitting number less than or equal to a fixed k is minor closed, which implies the existence of a corresponding polynomial-time recognition algorithm.

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