By Ding-Zhu Du, Peng-Jun Wan

ISBN-10: 1461452414

ISBN-13: 9781461452416

-Preface.-1. Introduction.-2. CDS usually Graph-3. CDS in Unit Disk Graph.-4. CDS in Unit Ball Graphs and development Bounded Graphs.-5. Weighted CDS in Unit Disk Graph.-6. Coverage.-7. Routing-Cost limited CDS.-8. CDS in Disk-Containment Graphs.-9. CDS in Disk-Intersection Graphs.-10. Geometric Hitting Set and Disk Cover.-11. Minimum-Latency Scheduling.-12 CDS in Planar Graphs.-Bibliography

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**Sample text**

Assume a1 , . . , a4 lie counter-clockwisely in disk1 (u) and a5 , . . , a8 lie counter-clockwisely in disk1 (v). Denote by ubi the radius passing through ai for i = 2, . . , 4 and by vbi the radius passing through ai for i = 5, . . , 8. Using arcs with radius one, draw four arc-triangles ub2 c2 , ub3 c3 , vb6 c6 , and vb7 c7 as shown in Fig. 11. Their boundaries intersect the boundary of disk1 (u) ∩ disk1 (v) at d2 , d3 , d6 , d7 , respectively. Note that none of a1 , a4 , a5 , a8 can lie in the four arc-triangles ub2 c2 , ub3c3 , vb6 c6 , and vb7 c7 .

4. 5 and with centers at nodes in a CDS. With this approach and Voronoi division, Funke et al. 291. However, in their proof, a key geometric extreme property was used without proof. Therefore, some researchers could not accept this result. Gao et al. [54] gave a detail proof of the geometric extreme property. Li et al. [72] improved approach of Funke et al. 8185. This is the best-known bound so far. 2 NP-Hardness and PTAS In this section, we give a new proof of NP-hardness and a new construction of PTAS for MIN-CDS in unit disk graphs.

4. Let A and B be two vertex subsets. If both G[B] and G[X] are connected, then −ΔX f (A ∪ B) + ΔX f (A) ≤ 1. Proof. Since q is submodular, we have ΔX q(A) ≤ ΔX q(A ∪ B). Moreover, since both subgraphs G[B] and G[X] are connected, the number of black components dominated by X in G[A ∪ B] is at most one more than the number of black components dominated by X in G[A]. Therefore, −ΔX p(A ∪ B) ≤ −ΔX p(A) + 1. Hence, −ΔX f (A ∪ B) ≤ −ΔX f (A) + 1. Let C∗ be a minimum CDS. We show two properties of C∗ in the following two lemmas.

### Connected dominating set : theory and applications by Ding-Zhu Du, Peng-Jun Wan

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