Algorithms and Turing kernels for detecting and counting small patterns in unit disk graphs

In this paper we investigate the parameterized complexity of counting and detecting small patterns in unit disk graphs: Given an n-vertex unit disk graph G with an embedding of ply p (i.e. G is an intersection graph of closed unit disks, and each point is contained in at most p disks) and a k-vertex unit disk graph P, count the number of (induced) copies of P in G. For general patterns P, we give an $2^{O(pk/\log k)}{n}^{O\left(1\right)}$ time algorithm for counting pattern occurrences. We show this is tight, even for ply $p=2$: any $2^{o(n/\log n)}{n}^{O\left(1\right)}$ time algorithm violates the Exponential Time Hypothesis (ETH). Our approach combines tools developed for planar subgraph isomorphism such as ‘efficient inclusion-exclusion’ from Nederlof (2020) [15], and ‘isomorphisms checks’ from Bodlaender et al. (2016) [5] with a different separator hierarchy and a new bound on the number of non-isomorphic separations tailored for unit disk graphs.