Algorithms and Turing Kernels for Detecting and Counting Small Patterns in Unit Disk Graphs

Abstract: In this paper we investigate the parameterized complexity of the task of counting and detecting occurrences of small patterns in unit disk graphs: Given an $n$-vertex unit disk graph $G$ with an embedding of ply $p$ (that is, the graph is represented as intersection graph with closed disks of unit size, 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 $2^{O(p k /\log k)}n<sup>{O(1)}$ time algorithm for counting pattern occurrences. We show this is tight, even for ply $p=2$ and $k=n$: any $2^{o(n/\log n)}n<sup>{O(1)}$ time algorithm violates the Exponential Time Hypothesis (ETH). For most natural classes of patterns, such as connected graphs and independent sets we present the following results: First, we give an $(pk)<sup>{O(\sqrt{pk})}n<sup>{O(1)}$ time algorithm, which is nearly tight under the ETH for bounded ply and many patterns. Second, for $p= k<sup>{O(1)}$ we provide a Turing kernelization (i.e. we give a polynomial time preprocessing algorithm to reduce the instance size to $k<sup>{O(1)}$). Our approach combines previous tools developed for planar subgraph isomorphism such as efficient inclusion-exclusion' from [Nederlof STOC'20], andisomorphisms checks' from [Bodlaender et al. ICALP'16] with a different separator hierarchy and a new bound on the number of non-isomorphic separations of small order tailored for unit disk graphs.