Premium
Absence of percolation in graphs based on stationary point processes with degrees bounded by two
Author(s) -
Jahnel Benedikt,
Tóbiás András
Publication year - 2023
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.21084
Subject(s) - mathematics , bounded function , point process , combinatorics , poisson point process , conjecture , connected component , discrete mathematics , graph , random graph , random geometric graph , mathematical analysis , line graph , voltage graph , statistics
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 bidirectionalk $$ k $$ ‐nearest neighbor graph of a two‐dimensional homogeneous Poisson point process does not percolate fork = 2 $$ k=2 $$ .