A Color-Avoiding Approach to Subgraph Counting in Bounded Expansion Classes

We present an algorithm to count the number of occurrences a pattern graph H on h vertices as induced subgraph in host G. If G belongs bounded expansion class, runs linear time, if nowhere dense class it almost-linear time. Our design choices are motivated by need for approach that can be engineered into practical implementation sparse graphs. Specifically, we introduce decomposition called counting dag \(\vec {C}(H)\) which encodes order-aware, inclusion-exclusion method H. Given such and suitable ordering \(\mathbb {G}\) input, our times appears time \(O(\Vert \vec {C}\Vert \cdot {\text {wcol}}_{h} (\mathbb {G})^{h-1} |G|)\), where \( {G})\) denotes maximum size weakly h-reachable sets {G}\). This implies, combined with previous results, running \(O((3h^2 (G))^{h^2} |G|)\) only takes input. note small modification, instead use strongly {col}}_{h} resulting overall complexity \(O(h (3 when given Because orderings weakly/strongly reachable computed relatively efficiently practice (Nadara et al.: J Exp Algorithmics 103:14:1–14:16, 2018), provides promising alternative algorithms using traditional p-treedepth coloring framework (O’Brien Sullivan in: Experimental evaluation isomorphisms classes expansion, CoRR, arXiv:1712.06690, 2017). describe preliminary experimental results from initial open source highlight its potential.