Touching Graphs of Unit Balls
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Abstract:
Abstract. The touching graph of balls is a graph that admits a representation by non-intersecting balls in the space (of prescribed dimension), so that its edges correspond to touching pairs of balls. By a classical result of Koebe [5], the disc touching graphs are exactly the planar graphs. This paper deals with a recognition of unit-ball touching graphs. The 2--dimensional case was proved to be NP-hard by Breu and Kirkpatrick [1]. We show in this paper that also unit-ball touching graphs in dimensions 3 and 4 are NP-hard to recognize. By a recent result of Kirkpatrick and Rote, these results may be transferred in ball-touching graphs in one dimension higher. 1
Citations
| 93 | Representation of a finite graph by a set of intervals on a real line – Lekkerkerker, Boland - 1962 |
| 70 | Kontaktprobleme der konformen Abbildung – Koebe - 1936 |
| 17 | On triangle contact graphs – Fraysseix, Mendez, et al. - 1994 |
| 8 | On the complexity of recognizing intersection and touching graphs of discs – Breu, Kirkpatrick - 1996 |
| 8 | Contact graphs of curves (extended abstract), in Graph Drawing (F.J.Brandenburg ed – Hlinen'y - 1996 |
| 3 | Intersection graphs of noncrossing arc-connected sets in the plane – Kratochv'il - 1996 |
