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 MAXIMUM PLANAR SUBGRAPH problem---given a graph G, find a largest planar subgraph of G---has applications in circuit layout, facility layout, and graph drawing. No previous polynomialtime approximation algorithm for this NP-Complete problem was known to achieve a performance ratio larger than 1=3, which is achieved simply by producing a spanning tree of G. We present the first approximation algorithm for MAXIMUM PLANAR SUBGRAPH with higher performance ratio (4=9 instead of 1=3). We also apply our algorithm to find large outerplanar subgraphs. Last, we show that both MAXIMUM PLANAR SUBGRAPH and its complement, the problem of removing as few edges as possible to leave a planar subgraph, are Max SNP-Hard.

A planarizing set of a graph is a set of edges or vertices whose removal leaves a planar graph. It is shown that, if G is an n-vertex graph of maximum degree d and orientable genus g, then there exists a planarizing set of O( p dgn) edges. This result is tight within a constant factor. Similar results are obtained for planarizing vertex sets and for graphs embedded on nonorientable surfaces. Planarizing edge and vertex sets can be found in O(n + g) time, if an embedding of G on a surface of genus g is given. We also construct an approximation algorithm that finds an O( p gn log g) planarizing vertex set of G in O(n log g) time if no genus-g embedding is given as an input. 1 Introduction A graph G is planar if G can be drawn in the plane so that no two edges intersect. Planar graphs arise naturally in many applications of graph theory, e.g. in VLSI and circuit design, in network design and analysis, in computer graphics, and is one of the most intensively studied class of graphs [2...

The problem of finding a maximum spanning planar subgraph of a nonplanar graph is NP-Complete. Several heuristics for the problem have been devised but their worst-case performance is unknown, although a trivial lower bound of 1/3 the optimum number of edges is easily shown. We discuss a new heuristic, based on spanning trees, for generating a subgraph with size at least 2/3 of the optimum for any input graph. The skewness of the n-dimensional hypercube Qn is also derived. Finally, we explore the relationship between the skewness and crossing number of a graph.

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.

Introduction The problem of extracting a maximum planar subgraph from a nonplanar graph, referred to as graph planarization, has important applications in circuit layout, facility layout, and automated graphical display systems [F, TDB]. The problem is NP-hard [LG]; hence, research has focused on heuristics. There are several algorithms for finding maximal planar subgraphs [CHT, CNS, GT, JTS, JM, K, OT]. However, there are graphs (see [CC]) for which the size ratio between two maximal planar subgraphs can be as small as 1=3. Hence, unless some precautions are taken to avoid the extraction of small subgraphs, these heuristics have the potential for poor behavior. In this paper, we analyze the worst-case performance of some heuristics and show that there are graphs which can cause each of them to achieve the 1=3 bound. However, a theoretical analysis of an algorithm's performance is often too pessimistic and somew

The problem of computing a maximal planar subgraph of a non-planar graph has been deeply investigated over the last 20 years. Several attempts have been tried to solve the problem with the help of PQ-trees. The latest attempt has been reported by Jayakumar et al. (1989). In this paper we show that the algorithm presented by Jayakumar et al. is not correct. We show that it does not necessarily compute a maximal planar subgraph and that the same holds for a modified version of the algorithm presented by Kant (1992). Our conclusions most likely suggest not to use PQ-trees at all for this specific problem.

We analyze several heuristics for graph planarization, i.e., deleting the minimum number of edges from a nonplanar graph to make it planar. The problem is NP-hard, although some heuristics which perform well in practice have been reported. In particular, we compare the two principle methods, based on path addition and vertex addition, respectively, with a selective edge addition method, an incremental method, and a "cycle packing" approach. For the incremental, the path addition, and the edge addition methods, we prove theoretical worst-case performance bounds of 1=3. We also present an empirical analysis of the heuristics. Our results indicate that the "cycle-packing" method consistently yields the best solutions when applied to a large set of test graphs. 1

The problem of computing a maximal planar subgraph of a non planar graph has been deeply investigated over the last 20 years. Several attempts have been tried to solve the problem with the help of PQ-trees. The latest attempt has been reported by Jayakumar et al. [10]. In this paper we show that the algorithm presented by Jayakumar et al. is not correct. We show that it does not necessarily compute a maximal planar subgraph and we note that the same holds for a modified version of the algorithm presented by Kant [12]. Our conclusions most likely suggest not to use PQ-trees at all for this specific 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