Bounding tree-width via contraction on the projective plane and torus (Q888611)

Summary: If \(X\) is a collection of edges in a graph \(G\), let \(G/X\) denote the contraction of \(X\). Following a question of \textit{J. Oxley} [private communication] and a conjecture of \textit{B. Oporowski} [unpublished], we prove that every projective planar graph \(G\) admits an edge partition \(\{X,Y\}\) such that \(G/X\) and \(G/Y\) have tree-width at most three. We prove that every toroidal graph \(G\) admits an edge partition \(\{X,Y\}\) such that \(G/X\) and \(G/Y\) have tree-width at most three and four, respectively.