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Recently, I am reading paper An Upper Bound for Resolution Size: Characterization of Tractable SAT Instances, which use tree decomposition to give an upper bound for SAT resolution refutation.

For a given nice tree decomposition $T=(V(T),G(T))$.

My question: Why $O(3^{treewidth})*V(T)$ is still $O(3^{treewidth})$

my question comes from the process of proving Theorem 3 from Lemma 2 enter image description here

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I don't see the claim that $O(3^{tw}) \cdot |V(T)| = O(3^{tw})$, as you mention.

Instead, $O^*(3^{tw}) \cdot |V(T)| = O^*(3^{tw})$. The notation $O^*$ is a shorthand used to suppress polynomial factors in the input size $n$. You should read $O^*(3^{tw})$ as $O(3^{tw} \cdot \text{poly}(n))$.

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