Absence of percolation in graphs based on stationary point processes with degrees bounded by two (extended abstract)

Jahnel, Benedikt and Tóbiás, András József (2023) Absence of percolation in graphs based on stationary point processes with degrees bounded by two (extended abstract). In: Proceedings of the 12th Japanese-Hungarian Symposium on Discrete Mathematics and Its Applications. BME VIK Számítástudományi és Információelméleti Tanszék, Budapest, pp. 537-542. ISBN 9789634219033

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Abstract

We consider undirected graphs that arise as deterministic functions of stationary point processes such that each point has degree bounded by two. For a large class of point processes and edge-drawing rules, we show that the arising graph has no infinite connected component, almost surely. In particular, this extends our previous result for signal-to-interference ratio graphs based on stabilizing Cox point processes and verifies the conjecture of Balister and Bollobás that the bidirectional k-nearest neighbor graph of a two-dimensional homogeneous Poisson point process does not percolate for k = 2.

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