We present an algorithm to count the number of occurrences of a pattern graph $H$ as an induced subgraph in a host graph $G$. If $G$ belongs to a bounded expansion class, the algorithm runs in linear time. Our design choices are motivated by the need for an approach that can be engineered into a practical implementation for sparse host graphs. Specifically, we introduce a decomposition of the pattern $H$ called a counting dag $\vec C(H)$ which encodes an order-aware, inclusion-exclusion counting method for $H$. Given such a counting dag and a suitable linear ordering $\mathbb G$ of $G$ as input, our algorithm can count the number of times $H$ appears as an induced subgraph in $G$ in time $O(\|\vec C\| \cdot h \text{wcol}_{h}(\mathbb G)^{h-1} |G|)$, where $\text{wcol}_h(\mathbb G)$ denotes the maximum size of the weakly $h$-reachable sets in $\mathbb G$. This implies, combined with previous results, an algorithm with running time $O(4^{h^2}h (\text{wcol}_h(G)+1)^{h^3} |G|)$ which only takes $H$ and $G$ as input. We note that with a small modification, our algorithm can instead use strongly $h$-reachable sets with running time $O(\|\vec C\| \cdot h \text{col}_{h}(\mathbb G)^{h-1} |G|)$, resulting in an overall complexity of $O(4^{h^2}h \text{col}_h(G)^{h^2} |G|)$ when only given $H$ and $G$. Because orderings with small weakly/strongly reachable sets can be computed relatively efficiently in practice [11], our algorithm provides a promising alternative to algorithms using the traditional $p$-treedepth colouring framework [13]. We describe preliminary experimental results from an initial open source implementation which highlight its potential.

中文翻译：

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一种有界扩展类中子图计数的避色方法

我们提出了一种算法来计算模式图 $H$ 作为宿主图 $G$ 中的诱导子图的出现次数。如果 $G$ 属于有界扩展类，则算法在线性时间内运行。我们的设计选择的动机是需要一种可以设计成稀疏宿主图的实际实现的方法。具体来说，我们引入了模式 $H$ 的分解，称为计数 dag $\vec C(H)$，它编码了 $H$ 的顺序感知、包含-排除计数方法。给定这样的计数 dag 和合适的线性排序 $G$ 的 $\mathbb G$ 作为输入，我们的算法可以计算 $H$ 在 $O(\|\ vec C\| \cdot h \text{wcol}_{h}(\mathbb G)^{h-1} |G|)$, 其中 $\text{wcol}_h(\mathbb G)$ 表示 $\mathbb G$ 中弱 $h$ 可达集的最大大小。这意味着，结合之前的结果，一个运行时间为 $O(4^{h^2}h (\text{wcol}_h(G)+1)^{h^3} |G|)$ 的算法只以 $H$ 和 $G$ 作为输入。我们注意到，通过一个小的修改，我们的算法可以使用强 $h$-reachable 集合，运行时间 $O(\|\vec C\| \cdot h \text{col}_{h}(\mathbb G) ^{h-1} |G|)$，导致 $O(4^{h^2}h \text{col}_h(G)^{h^2} |G|)$ 的整体复杂度，当仅给出 $H$ 和 $G$。因为在实践中可以相对有效地计算具有小的弱/强可达集的排序 [11]，我们的算法为使用传统 $p$-treedepth 着色框架 [13] 的算法提供了一种有前途的替代方案。