Optimization Problems in Unit-Disk Graphs

Unit-Disk Graphs (UDGs) are intersection graphs of equal diameter (or unit diameter w.l.o.g.) circles in the Euclidean plane. In the geometric (or disk) representation, each circle is specified by the coordinates of its center. Three equivalent graph models can be defined with vertices representing the circles [18]. In the intersection graph model, two vertices are adjacent if the corresponding circles intersect (tangent circles are also said to intersect). In the containment graph model, two vertices are adjacent when one circle contains the center of the other. In the proximity graph model, an edge exists between two vertices if the Euclidean distance between the centers of corresponding circles is within a specified bound. Recognizing UDGs is NP-hard [10] and hence no polynomial time algorithm is known for deriving the geometric representation from the graph model. From an algorithmic perspective this places an emphasis on whether or not the geometric representation is needed as input. UDGs are not necessarily perfect or planar [18] as several other geometric intersection graph classes are and thus motivate the need for dedicated theoretical study. The remainder of this article is organized as follows. We introduce the necessary def-