On Numbers of Simplicial Walks and Equivalent Canonizations for Graph Recognition

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 $k$.