I understand that treewidth and branchwidth are essentially equivalent for a fixed graph, given that $branchwidth(G) = Θ(treewidth(G))$.
However, my question pertains to a specific case involving Tseitin formulas.
The underlying graph of Tseitin formulas is denoted by $G$.
And the underlying hypergraph is denoted by $G'$ where each edge in the hypergraph corresponds to a clause of formulas.$G$ and $G'$ can be seen as dual graphs, although a vertex in $G$ may correspond to multiple edges in $G'$.
Myquestion: What's the connection between $branchwidth(G')$ and $treewidth(G)$
My question comes from two papers, in Characterizing Tseitin-formulas with short regular resolution refutations, it stated that:
"The upper bound for this result was already known from [2]where itis shown that, for every graph G, unsatisfiable Tseitin-formulas withthe underlying graph G have regular resolution refutations of lengthat most $2^{O(tw(G))}|V (G)|^c$ where c is a constant."
However, upon checking the reference, I noticed that it uses $branchwidth(G')$ rather than $treewidth(G)$.