We show that any nondeterministic read-once branching program that decides a satisfiable Tseitin formula based on an n× n grid graph has size at least 2^Ω(n). Then using the Excluded Grid Theorem by Robertson and Seymour we show that for arbitrary graph G(V, E) any nondeterministic read-once branching program that computes a satisfiable Tseitin formula based on G has size at least (formula presented) for all (formula presented)1/36, where tw (G) is the treewidth of G (for planar graphs and some other classes of graphs the statement holds for δ= 1). We also show an upper bound of (formula presented), where pw (G) is the pathwidth of G. We apply the mentioned results to the analysis of the complexity of derivations in the proof system OBDD (∧, reordering) and show that any OBDD (∧, reordering) -refutation of an unsatisfiable Tseitin formula based on a graph G has size at least (formula presented).