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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.

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...

In this paper we study on-line local routing algorithms for communication networks. Our algorithms take advantage of the geometric properties of planar networks. We pay special attention to on-line local routing algorithms which guarantee that a message reaches its destination. A message cosists of packets of data that have to be sent to a destination node, i.e. the message itself plus a finite amount of space used to record a constant amount of data to aid it in its traversal, e.g. the address of the starting and destination nodes, a constant number of nodes visited, etc. Local means that at each site we have at our disposal only local information regarding a node and its neighbors, i.e. no global knowledge of the network is available at any time, other that the network is planar and connected. We then develop location aided local routing algorithms for wireless communication networks, in particularly cellular telephone networks.

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In this paper we investigate the problem of identifying a planar subgraph of maximum weight of a given edge weighted graph. In the theoretical part of the paper, the polytope of all planar subgraphs of a graph G is defined and studied. All subgraphs of a graph G, which are subdivisions of K 5 or K 3;3 , turn out to define facets of this polytope. We also present computational experience with a branch and cut algorithm for the above problem. Our approach is based on an algorithm which searches for forbidden substructures in a graph that contains a subdivision of K 5 or K 3;3 . These structures give us inequalities which are used as cutting planes.

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.

We consider two graph invariants that are used as a measure of nonplanarity: the splitting number of a graph and the size of a maximum planar subgraph. The splitting number of a graph G is the smallest integer k 0, such that a planar graph can be obtained from G by k splitting operations. Such operation replaces a vertex v by two nonadjacent vertices v 1 and v 2 , and attaches the neighbors of v either to v 1 or to v 2 . We prove that the splitting number decision problem is NP-complete. We obtain as a consequence that planar subgraph remains NP-complete when restricted to graphs with maximum degree 3, when restricted to graphs with no subdivision of K 5 , or when restricted to cubic graphs, problems that have been open since 1979.

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