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Planarity Testing and Embedding
, 2004
"... Testing the planarity of a graph and possibly drawing it without intersections is one of the most fascinating and intriguing problems of the graph drawing and graph theory areas. Although the problem per se can be easily stated, and a complete characterization of planar graphs was available since 19 ..."
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Testing the planarity of a graph and possibly drawing it without intersections is one of the most fascinating and intriguing problems of the graph drawing and graph theory areas. Although the problem per se can be easily stated, and a complete characterization of planar graphs was available since 1930, an efficient solution to it was found only in the seventies of the last century. Planar graphs play an important role both in the graph theory and in the graph drawing areas. In fact, planar graphs have several interesting properties: for example they are sparse, fourcolorable, allow a number of operations to be performed efficiently, and their structure can be elegantly described by an SPQRtree (see Section 3.1.2). From the information visualization perspective, instead, as edge crossings turn out to be the main culprit for reducing readability, planar drawings of graphs are considered clear and comprehensible. As a matter of fact, the study of planarity has motivated much of the development of graph theory. In this chapter we review the number of alternative algorithms available in the literature for efficiently testing planarity and computing planar embeddings. Some of these algorithms
Crossing Number Bounds for the Twisted Cube
, 2001
"... The twisted cube TQ d is formed from the ddimensional hypercube Q d by `twisting' a pair of independent edges of any 4cycle of Q d . Asymptotic upper and lower bounds for the crossing numbers of the twisted cube and generalized twisted cube (GTQ d ) graphs are derived. We also determine the skewne ..."
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The twisted cube TQ d is formed from the ddimensional hypercube Q d by `twisting' a pair of independent edges of any 4cycle of Q d . Asymptotic upper and lower bounds for the crossing numbers of the twisted cube and generalized twisted cube (GTQ d ) graphs are derived. We also determine the skewness of TQ d . 1