Jörg Flum | Searcharxiv

arXiv2025

Some remarks on the uncolored versions of the original CFI-graphs

The CFI-graphs, named after Cai, Fürer, and Immerman, are central to the study of the graph isomorphism testing and of first-order logic with counting. They are colored graphs, and the coloring plays a role in many of their applications. As usual, it is not hard to remove the coloring by some extra graph gadgets, but at the cost of blowing up the size of the graphs and changing some parameters of them as well. This might lead to suboptimal combinatorial bounds important to their applications. Since then for some uncolored variants of the CFI-graphs it has been shown that they serve the same purposes. We show that this already applies to the graphs obtained from the original CFI-graphs by forgetting the colors. Moreover, we will see that there is a first-order formula $φ(x,y)$ expressing in almost all uncolored CFI-graphs that $x$ and $y$ have the same color in the corresponding colored graphs.

cs.DM↗

arXiv2023

On Algorithms Based on Finitely Many Homomorphism Counts

It is well known [Lovász, 67] that up to isomorphism a graph~$G$ is determined by the homomorphism counts $\hom(F, G)$, i.e., the number of homomorphisms from $F$ to $G$, where $F$ ranges over all graphs. Thus, in principle, we can answer any query concerning $G$ with only accessing the $\hom(\cdot,G)$'s instead of $G$ itself. In this paper, we deal with queries for which there is a hom algorithm, i.e., there are finitely many graphs $F_1, \ldots, F_k$ such that for any graph $G$ whether it is a Yes-instance of the query is already determined by the vector\[\overrightarrow{\hom}_{F_1,\ldots,F_k}(G):= \big(\hom(F_1,G),\ldots,\hom(F_k,G)\big),\]where the graphs $F_1, \ldots, F_k$ only depend on $φ$. We observe that planarity of graphs and 3-colorability of graphs, properties expressible in monadic second-order logic, have no hom algorithm. On the other hand, queries expressible as a Boolean combination of universal sentences in first-order logic FO have a hom algorithm. Even though it is not easy to find FO definable queries without a hom algorithm, we succeed to show this for the non-existence of an isolated vertex, a property expressible by the FO sentence $\forall x\exists y Exy$, somehow the ``simplest'' graph property not definable by a Boolean combination of universal sentences.These results provide a characterization of the prefix classes of first-order logic with the property that each query definable by a sentence of the prefix class has a hom algorithm. For adaptive query algorithms, i.e., algorithms that again access $\overrightarrow{\hom}_{F_1,\ldots,F_k}(G)$ but here $F_{i+1}$ might depend on $\hom(F_1,G),\ldots,\hom(F_i,G)$, we show that three homomorphism counts $\hom(\cdot,G)$ are both sufficient and in general necessary to determine the isomorphism type of $G$.

cs.CC↗