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TR21-097 | 7th July 2021 15:34
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#### Number of Variables for Graph Identification and the Resolution of GI Formulas

**Abstract:**
We show that the number of variables and the quantifier depth needed to distinguish a pair of graphs by first-order logic sentences exactly match the complexity measures of clause width and positive depth needed to refute the corresponding graph isomorphism formula in propositional narrow resolution.

Using this connection, we obtain upper and lower bounds for refuting graph isomorphism formulas in (normal) resolution. In particular, we show that if $k$ is the minimum number of variables needed to distinguish two graphs with $n$ vertices each, then there is an $n^{\mathrm{O}(k)}$ resolution refutation size upper bound for the corresponding isomorphism formula, as well as lower bounds of $2^{k-1}$ and $k$ for the tree-like resolution size and resolution clause space for this formula. We also show a resolution size lower bound of ${\exp} \big( \Omega(k^2/n) \big)$ for the case of colored graphs with constant color class size.

Applying these results, we prove the first exponential lower bound for graph isomorphism formulas in the proof system SRC-1, a system that extends resolution with a global symmetry rule, thereby answering an open question posed by Schweitzer and Seebach.