Graph minors and graphs on surfaces (Q2741176)

The author reviews the recent new development on graph minors and graphs on surfaces. Special attention is paid to the now classical excluded minor theorem by Robertson and Seymour: For every surface \(S\), there exists a finite class of graphs \(\text{Forb}(S)\) such that a graph \(G\) can be embedded on \(S\) if and only if no minor of \(G\) is in \(\text{Forb}(S)\). In 1997 the reviewer published a short proof of this theorem; see \textit{C. Thomassen} [J. Comb. Theory, Ser. B 70, No. 2, 306-311 (1997; Zbl 0882.05053)]. However, that proof depends on two deep theorems in the Robertson-Seymour theory: (1) every graph of large tree-width contains a large grid; (2) The graphs in \(\text{Forb}(S)\) have bounded tree-width. In 1999 \textit{R. Diestel, T. R. Jensen, K. Yu. Gorbunov,} and \textit{C. Thomassen} [J. Comb. Theory, Ser. B 75, No. 1, 61-73 (1999; Zbl 0949.05075)] published a resonably short proof of (1). In the present paper, Mohar presents a short proof of (2).NEWLINENEWLINEFor the entire collection see [Zbl 0964.00035].