Transfer of Weisfeiler-Leman equivalence under graph operations

The Weisfeiler-Leman algorithms WL(k) for $k\ge1$ form a hierarchy of increasingly powerful tests for graph isomorphism. We study how the equivalence relations they induce on graphs transfer through a graph operation $Φ$: for a given $k$, what level of WL-equivalence between graphs $H_1$ and $H_2$ guarantees WL(k)-equivalence of graphs $Φ(H_1)$ and $Φ(H_2)$? We call the least such level the transfer parameter $τ_Φ(k)$ and determine it exactly for the line graph operation $Φ= L$: $τ_L(k)=2k+1$ for $k\ge2$, while $τ_L(1)=1$. We also extend these results to the $(r,S)$-line graphs, which generalize the line graph from edges to $r$-cliques by connecting two cliques when their intersection size lies in a prescribed set $S$. For these operations we prove that $τ_{r,S}(1)=r-1$ for every $S$ and, for certain choices of $S$, that $τ_{r,S}(k)=rk+r-1$ for $k\ge r$, and we bound the transfer parameter in the remaining range. Our upper bounds on transfer parameters follow from a treewidth analysis of diagrammatic expansions of homomorphism counts into $Φ(H)$ as linear combinations of homomorphism counts into $H$. The matching lower bounds combine explicit constructions of witness graphs with the homomorphism-distinguishing closedness of graphs of bounded treewidth.