By Siceloff L.P., Wentworth G., Smith D.E.

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The size of a disk is its radius and the size of a square is its side length. 3 If a (unit) disk graphs has a q-representation for some q ≥ 0, then it has a (4q + 6)-separated representation. Proof: Let G be a (unit) disk graph and S a q-representation for G. Scale the space by 2q such that all numbers of S are 2q-bit integers. We claim that any two nontouching disks in S are δ-separated, where δ = 1/(24q+4 ). For any u ∈ S, let cu = (xu , yu ) denote the center and ru the radius of disk u. Suppose two disks u and v intersect but do not touch.

The theorem follows. 40 Chapter 4. 14 If the class of disk graphs has a q-separated representation for some q : N → N, then it has a q -representation, where q (n) = n(q(n) + log n + 3). In particular, polynomial separation implies polynomial representation. The same results hold, mutatis mutandis, for square graphs. Moreover, the theorems apply both to open and closed disks or squares. We can now prove the following result. 15 The class of intersection graphs of closed (unit) disks has a polynomial representation if and only if the class of intersection graphs of open (unit) disks has a polynomial representation.

Approximation on Geometric Intersection Graphs A dominating set in a wireless communication network can be seen as a set of emergency transmitters capable of reaching every node in the network, or as central nodes in node clusters. A connected dominating set can be used as a backbone for easier and faster communications. e. we look for (connected) dominating sets of minimum cardinality. Previous Work All problems mentioned above are NP-complete on general graphs (see Garey and Johnson [115]). Since (unit) disk graphs are a restricted class of graphs with a nice geometric interpretation, one might hope that these problems are better solvable.