Two graphs are isomorphic exactly when they admit the same number of homomorphisms from every graph. Hence, a graph is recognized up to isomorphism by homomorphism counts over the class of all graphs. Restricting to a specific graph class yields some natural isomorphism relaxations and modulates recognition to particular graph properties. A notable restriction is to the classes of bounded treewidth, yielding the isomorphism relaxation of Weisfeiler--Leman refinement (WL), as shown by Dvořák [JGT 2010]. The properties recognized by WL are exactly those definable in fragments of first-order logic with counting quantifiers, as shown by Cai, Fürer, and Immerman [Comb. 1992]. We characterize the restriction to the classes of bounded pathwidth by numbers of simplicial walks, and formalize it into a refinement procedure (SW). The properties recognized by SW are exactly those definable in fragments of restricted-conjunction first-order logic with counting quantifiers, introduced by Montacute and Shah [LMCS 2024]. Unlike WL, computing SW directly is not polynomial-time in general. We address this by representing SW in terms of multiplicity automata. We equip these automata with an involution, simplifying the canonization to standard forward reduction and omitting the backward one. The resulting canonical form is computable in time $O(kn^{3k})$ for any graph on $n$ vertices and the restriction to pathwidth at most $k$.

翻译：当且仅当两个图从每个图接收的同态数量相同时，它们才是同构的。因此，图在同构意义下可通过在所有图类上的同态计数来识别。将计数限制在特定图类中会产生一些自然的同构松弛，并将识别调整到特定的图性质。一个显著的约束是限制在有界树宽图类上，这产生了Weisfeiler--Leman细化（WL）的同构松弛，如Dvořák [JGT 2010]所示。WL识别的性质正是那些可在带计数量词的一阶逻辑片段中定义的，如Cai、Fürer和Immerman [Comb. 1992]所示。我们通过单纯行走的数量来刻画限制在有界路径宽图类上的情况，并将其形式化为一个细化过程（SW）。SW识别的性质正是那些在带计数量词的受限合取一阶逻辑片段中定义的，该逻辑由Montacute和Shah [LMCS 2024]引入。与WL不同，直接计算SW通常不是多项式时间的。我们通过用多重性自动机表示SW来解决这个问题。我们为这些自动机配备了一个对合，从而将正则化简化为标准的前向约简，并省略了后向约简。对于任何具有$n$个顶点且路径宽最多为$k$的图，所得的正则形式可在$O(kn^{3k})$时间内计算。