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.

We introduce the visibility number b(G) of a graph G, which is the minimum t such that G can be represented by assigning each vertex a union of at most t horizontal segments in the plane so that vertices u; v are adjacent if and only if some point assigned to u sees some point assigned to v via a vertical segment unobstructed by other assigned points. We prove the following: 1) every planar graph has visibility number at most 2, which is sharp. 2) r b(K m;n ) r + 1, where r = d(mn + 4)=(2m + 2n)e. 3) dn=6e b(K n ) dn=6e + 1. 4) When G has n vertices, b(G) dn=6e + 2.

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

this paper we give a survey of the topics and results in topological graph theory. We offer neither breadth, as there are numerous areas left unexamined, nor depth, as no area is completely explored. Nevertheless, we do offer some of the favorite topics of the author and attempt to place them 1

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).

AND ITS APPLICATION TO BIOMEDICAL LITERATURE F. Lunin . . . . . . 47 I

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