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.

We give a state-of-the-art survey of the thickness of a graph from both a theoretical and a practical point of view. After summarizing the relevant results concerning this topological invariant of a graph, we deal with practical computation of the thickness. We present some modifications of a basic heuristic and investigate their usefulness for evaluating the thickness and determining a decomposition of a graph in planar subgraphs. Key words: Thickness, maximum planar subgraph, branch and cut 1 Introduction In VLSI circuit design, a chip is represented as a hypergraph consisting of nodes corresponding to macrocells and of hyperedges corresponding to the nets connecting the cells. A chip-designer has to place the macrocells on a printed circuit board (which usually consists of superimposed layers), according to several designing rules. One of these requirements is to avoid crossings, since crossings lead to undesirable signals. It is therefore desirable to find ways to handle wi...

Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215–221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.

The thickness problem on graphs is NP-hard and only few results concerning this graph invariant are known. Using a decomposition theorem of Truemper, we show that the thickness of the class of graphs without G 12 - minors is less than or equal to two (and therefore, the same is true for the more well-known class of the graphs without K 5 -minors). Consequently, the thickness of this class of graphs can be determined with a planarity testing algorithm in linear time.

this paper we consider a variation of geometric thickness which lies between thickness and geometric thickness in which each edge has at most one bend. We are also interested in drawings with small area, which is an important consideration in VLSI and visualisation. To measure the area of a drawing we assume a vertex resolution rule; that is, pairs of vertices are at least unit-distance apart. A drawing obtained from a book embedding by positioning the vertices around a circle, as discussed above, has O(n ) area. The construction in [DEH00] demonstrating that (Kn ) d 4 e has O(n 6 ) area [D. Eppstein, personal communication ]. We prove the following 2-dimensional generalisation of the above-mentioned result in [Mal94b] for producing book embeddings

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

The thickness of a graph is the minimum number of planar subgraphs into which the graph can be decomposed. Determining the thickness of a given graph is known to be a NP-complete problem. This paper discusses the possibility of determining the thickness of a graph by a genetic algorithm. Our tests show that the genetic approach outperforms the earlier heuristic algorithms reported in the literature.

The thickness of a graph is the minimum number of planar subgraphs into which the graph can be decomposed. This note discusses some recent attempts to determine upper bounds for the thickness of a graph as a function of the number of edges or as a function of its maximum degree.

The geometric thickness of a graph is the minimum number of layers such that the graph can be drawn in the plane with edges as straight-line segments, and with each edge assigned to a layer so that no two edges on the same layer cross. We consider a variation on this theme in which each edge is allowed one bend. We prove that the vertices of an n-vertex m-edge graph can be positioned in a $$ grid and the edges assigned to $$ layers, so that each edge is drawn with at most one bend and no two edges on the same layer cross. The proof is a 2-dimensional generalization of a theorem of S. M. Malitz [J. Algorithms 17(1):71-84, 1994] on book embeddings. We obtain a Las Vegas algorithm to compute the drawing in O(m log n log log n) time (with high probability).

The thickness problem on graphs is NP-hard and only few results concerning this graph invariant are known. Using decomposition theorems of Wagner and Truemper, we show that the thickness of graphs without K 5 - minors is less than or equal to two. Therefore, the thickness of this class of graphs can be determined with a planarity testing algorithm in linear time. Key words: Thickness, crossing number, skewness, graph-minor, 2-sum, \Delta-sum 1 Introduction The thickness `(G) of a graph G = (V; E) is the minimum number k such that G is the union of k planar subgraphs (here, by "union of k planar subgraphs" we mean that the edge-set E can be partitioned into k sets so that the graph induced by each set is planar). Therefore, the thickness is one measure of the degree of nonplanarity of a graph. Clearly, `(G) = 1 if and only if G is planar. The thickness problem, asking for the thickness of a given graph G, is NP-hard ([Man83]), so there is little hope to find a polynomial time ...