Cover Contact Graphs
Atienza, Nieves and De Castro, Natalia and Cortés, Carmen and Garrido, M. Ángeles M. Ángeles and Grima, Clara I. and Hernández, Gregorio and Márquez, Alberto and Moreno, Auxiliadora and Nöllenburg, Martin and Portillo, José Ramon and Reyes, Pedro and Valenzuela, Jesús and Villar, Maria Trinidad and Wolff, Alexander (2008) Cover Contact Graphs. In: Graph Drawing 15th International Symposium, GD 2007, September 24-26, 2007, Sydney, Australia , pp. 171-182 (Official URL: http://dx.doi.org/10.1007/978-3-540-77537-9_18).
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We study problems that arise in the context of covering certain geometric objects (so-called $seeds$, e.g., points or disks) by a set of other geometric objects (a so-called $cover$, e.g., a set of disks or homothetic triangles). We insist that the interiors of the seeds and the cover elements are pairwise disjoint, but they can touch. We call the contact graph of a cover a $cover contact graph$ (CCG). We are interested in two types of tasks: (a) deciding whether a given seed set has a connected CCG, and (b) deciding whether a given graph has a realization as a CCG on a given seed set. Concerning task (a) we give efficient algorithms for the case that seeds are points and covers are disks or triangles. We show that the problem becomes NP-hard if seeds and covers are disks. Concerning task (b) we show that it is even NP-hard for point seeds and disk covers (given a fixed correspondence between vertices and seeds).
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