Fully Dynamic Four-Vertex Subgraph Counting
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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 any four-vertex subgraph, which is not a clique, in deterministic amortized update time 𝒪(m^1/2), resp., 𝒪(m^2/3). Queries can be answered in constant time. For length-3 paths, paws, 4-cycles, and diamonds these 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 length-3 paths, 4-cycles, diamonds, or 4-cliques takes amortized update time Ω(m^1/2-δ). Additionally, for 4-cliques and all connected induced subgraphs, we show a lower bound of Ω(m^1-δ) for any small constant δ > 0 for the amortized update time, assuming the static combinatorial 4-clique conjecture holds. This shows that the 𝒪(m) algorithm by Eppstein et al. [9] for these subgraphs cannot be improved by a polynomial factor.
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