CDS in Disk-Containment Graphs

Abstract

Disk-containment graphs are generalizations of the unit disk graphs. Consider a finite planar set V of nodes. Each node v is associated with a disk of radius r v centered at v. The disk-containment graph (DCG) of V is the undirected graph \(G = \left (V,E\right )\) in which uv ∈ E if and only if the disk-associated u contains v and disk-associated v contains u. In other words, uv ∈ E if and only if the Euclidean distance between u and v is no more than \(\min \left \{{r}_{u},{r}_{v}\right \}\). When all the disks associated with the nodes in V have unit radius, then the DCG of V is exactly the UDG of V. The DCG arises naturally from communication topologies of multihop wireless networks with disparate communication ranges [102, 124]. Indeed, if V represents the set of nodes in a multihop wireless network and each r v represents the communication radius of the node v, the DCG of V is exactly the symmetric communication topology of the multihop wireless network.