Fully Dynamic Four-Vertex Subgraph Counting

This paper presents a comprehensive study of algorithms for maintaining the number of all connected four-vertex subgraphs in a dynamic graph. Specifically, our algorithms maintain the number of paths of length three in deterministic amortized $\mathcal{O}(m^\frac{1}{2})$ update time, and any other connected four-vertex subgraph which is not a clique in deterministic amortized update time $\mathcal{O}(m^\frac{2}{3})$. Queries can be answered in constant time. We also study the query times for subgraphs containing an arbitrary edge that is supplied only with the query as well as the case where only subgraphs containing a vertex $s$ that is fixed beforehand are considered. For length-3 paths, paws, $4$-cycles, and diamonds our bounds match or are not far from (conditional) lower bounds: Based on the OMv conjecture we show that any dynamic algorithm that detects the existence of paws, diamonds, or $4$-cycles or that counts length-$3$ paths takes update time $\Omega(m^{1/2-\delta})$. Additionally, for $4$-cliques and all connected induced subgraphs, we show a lower bound of $\Omega(m^{1-\delta})$ for any small constant $\delta > 0$ for the amortized update time, assuming the static combinatorial $4$-clique conjecture holds. This shows that the $\mathcal{O}(m)$ algorithm by Eppstein at al. for these subgraphs cannot be improved by a polynomial factor.