A point set S⊆ R2 is universal for a class G of planar graphs if every graph of G has a planar straight-line embedding on S. It is well-known that the integer grid is a quadratic-size universal point set for planar graphs, while the existence of a subquadratic universal point set still remains one of the most fascinating open problems in Graph Drawing. In this paper we make a major step towards a solution for this problem. Motivated by the fact that each point set of size n in general position is universal for the class of n-vertex outerplanar graphs, we concentrate our attention on k-outerplanar graphs. We prove that they admit an O(nlog n) -size universal point set in two distinct cases, namely when k= 2 (2-outerplanar graphs) and when k is unbounded but each outerplanarity level is restricted to be a simple cycle (simply-nested graphs).
Angelini, P., Bruckdorfer, T., Di Battista, G., Kaufmann, M., Mchedlidze, T., Roselli, V., et al. (2018). Small Universal Point Sets for k-Outerplanar Graphs. DISCRETE & COMPUTATIONAL GEOMETRY, 60(2), 430-470 [10.1007/s00454-018-0009-x].
Small Universal Point Sets for k-Outerplanar Graphs
Di Battista G.;Roselli V.;Squarcella C.
2018-01-01
Abstract
A point set S⊆ R2 is universal for a class G of planar graphs if every graph of G has a planar straight-line embedding on S. It is well-known that the integer grid is a quadratic-size universal point set for planar graphs, while the existence of a subquadratic universal point set still remains one of the most fascinating open problems in Graph Drawing. In this paper we make a major step towards a solution for this problem. Motivated by the fact that each point set of size n in general position is universal for the class of n-vertex outerplanar graphs, we concentrate our attention on k-outerplanar graphs. We prove that they admit an O(nlog n) -size universal point set in two distinct cases, namely when k= 2 (2-outerplanar graphs) and when k is unbounded but each outerplanarity level is restricted to be a simple cycle (simply-nested graphs).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.