We study the parameterized complexity of the problem of counting graph homomorphisms with given partial injectivity constraints, i.e., inequalities between pairs of vertices, which subsumes counting of graph homomorphisms, subgraph counting and, more generally, counting of answers to equi-join queries with inequalities. Our main result presents an exhaustive complexity classification for the problem in fixed-parameter tractable and \(\#\mathsf {W[1]}\)-complete cases. The proof relies on the framework of linear combinations of homomorphisms as independently discovered by Chen and Mengel (PODS 16) and by Curticapean, Dell and Marx in the recent breakthrough result regarding the exact complexity of the subgraph counting problem (STOC 17). Moreover, we invoke Rota’s NBC-Theorem to obtain an explicit criterion for fixed-parameter tractability based on treewidth. The abstract classification theorem is then applied to the problem of counting locally injective graph homomorphisms from small pattern graphs to large target graphs. As a consequence, we are able to fully classify its parameterized complexity depending on the class of allowed pattern graphs.

我们研究了在给定部分内射约束的情况下对图同态进行计数的问题的参数化复杂性，即，成对的顶点之间的不等式，这包括对图同态进行计数，对子图进行计数，并且更普遍地，对不等式的等联接查询的答案进行计数。我们的主要结果为固定参数可处理和\（\＃\ mathsf {W [1]} \）中的问题提供了详尽的复杂度分类-完整的案例。该证明依赖于Chen和Mengel（PODS 16）以及Curticapean，Dell和Marx在最近的关于子图计数问题的确切复杂性的突破性结果（STOC 17）中独立发现的同态线性组合框架。此外，我们调用Rota的NBC定理，以基于树宽获得用于固定参数可处理性的显式准则。然后将抽象分类定理应用于对从小模式图到大目标图的局部内射图同态进行计数的问题。因此，我们能够根据允许的模式图的类别将其参数化复杂度完全分类。