informs ® doi 10.1287/ijoc.1040.0074 © 2005 INFORMS This is the second of two papers dealing with the relationship of branchwidth and planar graphs. Branchwidth and branch decompositions, introduced by Robertson and Seymour, have been shown to be beneficial for both proving theoretical results on graphs and solving NP-hard problems modeled on graphs. The first practical implementation of an algorithm of Seymour and Thomas for computing optimal branch decompositions of planar hypergraphs is presented. This algorithm encompasses another algorithm of Seymour and Thomas for computing the branchwidth of any planar hypergraph, whose implementation is discussed in the first paper. The implementation also includes the addition of a heuristic to decrease the run times of the algorithm. This method, called the cycle method, is an improvement on the algorithm by using a “divide-and-conquer” approach. Key words: planar graph; branchwidth; branch decomposition; carvingwidth

This paper presents a novel protocol for a spatiotemporal variant of multicast called mobicast, designed to support message delivery in ad hoc sensor networks. The spatiotemporal character of mobicast relates to the obligation to deliver a message to all the nodes that will be present at time t in some geographic zone Z, where both the location and shape of the delivery zone are a function of time over some interval (tstart , t end ). The protocol, called Face-Aware Routing (FAR), exploits ideas adapted from existing applications of face routing to achieve reliable mobicast delivery. The key features of the protocol are a routing strategy, which uses information confined solely to a node's immediate spatial neighborhood, and a forwarding schedule, which employs only local topological information. Statistical results shows that, in uniformly distributed random disk graphs, the spatial neighborhood size is usually less than 20. This suggests that FAR is likely to exhibit a low average memory cost. An estimation formula for the average size of the spatial neighborhood in a random network is another analytical result reported in this paper. This paper also presents a novel and low cost distributed algorithm for spatial neighborhood discovery.

Abstract. In this paper we consider the problem of simultaneous drawing of two graphs. The goal is to produce aesthetically pleasing drawings for the two graphs by means of a heuristic algorithm and with human assistance. Our implementation uses the DiamondTouch table, a multiuser, touch-sensitive input device, to take advantage of direct physical interaction of several users working collaboratively. The system can be downloaded at

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Abstract — When exploring computing elements made from technologies other than CMOS, it is imperative to investigate the effects of physical implementation constraints. This paper focuses on molecular Quantum-dot Cellular Automata circuits. For these circuits, it is very difficult for chemists to fabricate wire crossings (at least in the near future). A novel technique is introduced to remove wire crossings in a given circuit to facilitate the self assembly of real circuits – thus providing meaningful and functional design targets for both physical and computer scientists. The technique eliminates all wire crossings with minimal logic gate/node duplications. Experimental results based on existing QCA circuits and other benchmarks are quite encouraging, and suggest that further investigation is needed. I.

Pach and Tóth [15] proved that any n-vertex graph of genus g and maximum degree d has a planar crossing number at most c g dn, for a constant c>1. We improve on this results by decreasing the bound to O(dgn), if g = o(n), and to O(g²), otherwise, and also prove that our result is tight within a constant factor.

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.

The splitting number of a graph G consists in the smallest positive integer k 0, such that a planar graph can be obtained from G by k splitting operations, such operation replaces v by two nonadjacent vertices v1 and v2 , and attaches the neighbors of v either to v1 or to v2 . One of the most useful graphs in computer science is the n--cube. Dean and Richter devoted an article to proving that the minimum number of crossings in an optimum drawing of the 4--cube is 8, but no results about splitting number of nonplanar n--cubes are known. In this note we give a proof that the splitting number of the 4--cube is 4. In addition, we give the lower bound 2 n\Gamma2 for the splitting number of the n--cube. In particular, because it is known that the splitting number of the n--cube is O(2 n ), our result implies that the splitting number of the n-cube is \Theta(2 n ).

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 What is a Planar Graph? . . . . . . . . . . . . . . . . . . . . 7 1.3 Planarity Testing . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.1 The Algorithm of Hopcroft and Tarjan . . . . . . . . . . . . . 10 1.3.2 The Algorithm of Lempel, Even and Cederbaum . . . . . . . 11 1.4 How to Make a Graph Planar . . . . . . . . . . . . . . . . . . 13 1.4.1 Inserting Vertices . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.2 Deleting Edges . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.5 How to Make a Planar Graph 2-Connected Planar . . . . . . 16 1.6 Convex Representations . . . . . . . . . . . . . . . . . . . . . 18 1.7 Methods Based on Canonical Orderings . . . . . . . . . . . . 22 1.7.1 The Algorithm of De Fraysseix, Pach and Pollack . . . . . . . 23 1.7.2 The Barycentric Algorithm of Schnyder . . . . . . . . . . . . 24 1.7.3 The Straight-Line Algorithm of Kant . . . . . . . . . . . . . . 26 1.7.

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