On Graphs Supported by Line Sets
Dujmovic, Vida and Evans, William and Kobourov, Stephen G. and Liotta, Giuseppe and Weibel, Christophe and Wismath, Stephen (2011) On Graphs Supported by Line Sets. In: Graph Drawing 18th International Symposium, GD 2010, September 21-24, 2010, Konstanz, Germany, , pp. 177-182 (Official URL: http://dx.doi.org/10.1007/978-3-642-18469-7_16).
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For a set S of n lines labeled from 1 to n, we say that S supports an n-vertex planar graph G if for every labeling from 1 to n of its vertices, G has a straight-line crossing-free drawing with each vertex drawn as a point on its associated line. It is known from previous work  that no set of n parallel lines supports all n-vertex planar graphs. We show that intersecting lines, even if they intersect at a common point, are more powerful than a set of parallel lines. In particular, we prove that every such set of lines supports outerpaths, lobsters, and squids, none of which are supported by any set of parallel lines. On the negative side, we prove that no set of n lines that intersect in a common point supports all n-vertex planar graphs. Finally, we show that there exists a set of n lines in general position that does not support all n-vertex planar graphs.
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