# Read e-book online Optimization and approximation on systems of geometric PDF

By van Leeuwen E.

ISBN-10: 9090243178

ISBN-13: 9789090243177

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Problems We consider various optimization problems on graphs that are relevant to geometric intersection graph models and specifically to (unit) disk graphs models of wireless communication networks. 1 Let G be a graph. A set S ⊆ V (G) is an independent set if there are no u, v ∈ S such that (u, v) ∈ E(G). A set S ⊆ V (G) is a vertex cover if for each (u, v) ∈ E(G) it holds that u ∈ S or v ∈ S. Observe that an independent set is the complement of a vertex cover (and vice versa) [115]. Furthermore, we are usually looking for a maximum independent set and a minimum vertex cover.

215] give a polynomial-time algorithm yielding √ a mapping of quality O((log5/2 n) · log log n). This clearly leaves a large gap and a major open question. 2. Disk Graphs and Ball Graphs 23 ball contact graphs. In two dimensions, ball contact graphs are also called disk contact graphs or coin graphs [231]. 1). Hence these graphs are recognizable in linear time [152]. If however the ratio of the radii of the largest and smallest disk is any fixed constant, then the recognition problem becomes NP-hard [41].

Recognizing whether the boxicity is at most d is NP-complete for any fixed d ≥ 2 [173, 273, 196]. Note that for d = 1, the recognition problem is equal to the problem of recognizing interval graphs, which is in P. Given the above, it is not hard to imagine a natural generalization of unit interval graphs. This leads to intersection graphs of d-dimensional axis-parallel unit cubes, which are d-dimensional axis-parallel boxes with side length equal to one. For d = 2, these are called unit square intersection graphs, or simply unit square graphs.