Abstract
A graph G is a Generalized Disk Graph if for some dimension \(\eta \ge 1\), a non-decreasing sub-linear function f and natural number t, each vertex \(v_i\) can be assigned a length \(l_i\) and set \(P_i \subseteq \mathbb {R}^\eta \) of t points such that \(v_iv_j\) is an edge of G if and only if \(l_i \le l_j\) and \(d(P_i, P_j) \le l_if(l_j/l_i)\,+\,l_if(1)\), where \(d(\cdot , \cdot )\) is the least distance between points in either set. Generalized disk graphs were introduced as a model of wireless network interference and have been shown to be dramatically more accurate than disk graphs or other previously known graph classes. However, their properties have not been studied extensively before.
We give a geometric representation of these graphs as intersection graphs of convex shapes, relate them to other geometric intersection graph classes, and solve several important optimization problems on these graphs using the geometric representation; either exactly (in two-dimensions) or approximately (in higher dimensions).
Work supported by grant 174484-051 from Icelandic Research Fund and grant 208348-0091 from Icelandic Student Innovation Fund.
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Notes
- 1.
We sometimes consider \(f(x)=x\) as an example; it is not sublinear but it satisfies all important properties we need (e.g., it is grounded w.r.t. hyperplane \(h=1\)).
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Arnþórsson, Í.M., Chaplick, S., Gylfason, J.S., Halldórsson, M.M., Reynisson, J.M., Tonoyan, T. (2021). Generalized Disk Graphs. In: Lubiw, A., Salavatipour, M., He, M. (eds) Algorithms and Data Structures. WADS 2021. Lecture Notes in Computer Science(), vol 12808. Springer, Cham. https://doi.org/10.1007/978-3-030-83508-8_9
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