IntroductionA unit disk graph is the intersection graph of closed disks of diameter 1 in the plane. It has been widely studied from bothpractical and theoretical points of view. Unit disk graphs representable in a limited area are also considered in the literature.Ito and Kadoshita [5] showed that the maximum independent set problem and the minimum dominating set problem areboth W[1]-complete for unit disk graphs with all centers lying in a square of side length√t, when parameterized by thearea t. Breu [2] studied the unit disk graphs whose centers are in the area {(x, y) : −∞ < x < ∞, 0 ≤ y ≤ c}. Such unitdisk graphs are called c-strip graphs. Breu showed that every√3/2-strip graph is a co-comparability graph [2, Section 3.1.2].We denote the class of unit disk graphs by UDG, and the class of c-strip graphs by SG(c). A unit interval graph is the intersectiongraph of closed intervals of length 1 in the real line. We denote the class of unit interval graphs by UIG. From thedefinitions, it follows that UIG = SG(0) ⊆ UDG. It is easy to show that indeed the inclusion is proper. For convenience sake,let SG(∞) = UDG.In this paper, we consider the graphs that are c-strip graphs for every c > 0, which can be expressed in our notation as0appear that TSG = SG(0). However, it can be seen that SG(0) = UIG ( TSG from the following observations. From thedefinitions, SG(0) ⊆ TSG holds. It is known that K1,3 ̸∈ UIG [10]. On the other hand, K1,3 ∈ TSG, because the centers(−1, 0), (0, 0), (1, 0), and (0, ε) of unit disks represent K1,3 for every 0 < ε ≤ 1.
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