By Ding-Zhu Du, Peng-Jun Wan

-Preface.-1. Introduction.-2. CDS quite often Graph-3. CDS in Unit Disk Graph.-4. CDS in Unit Ball Graphs and progress 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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**Example text**

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.

Every subset of the maximal independent set constructed as above is within distance two from its complement. In the second stage, consider the constructed maximal independent set as a set of terminals and then find the minimum number of Steiner nodes (added nodes) to interconnect all terminal. This means to solve the following problem. 3 Two-Stage Algorithm 45 ST-MSP-IN -UDG: Given a unit disk graph G = (V, E) and a node subset P ⊆ V with AoA Property, find a node subset S with the minimum cardinality, such that G[P∪S] is connected.

Pk are k shortest paths from u to k different orphan heads in C. We consider S = p1 ∪ · · · ∪ pk as a spider although p1 , . . , pk may have some common nodes other than u. When calculate cost(S), we assume that all p1 , . . , pk are disjoint except at body node. Therefore, cost(S) is actually an upper bound for the total weight of increased internal nodes. The purpose to make this assumption is to have an easy way to compute quotient(v) for every node u, which is defined to be quotient(u) = min{quotient(S) | S is over all legal spider with body node u}.