Birational morphisms of the plane

Let $A^2$ be the affine plane over a field $K$ of characteristic $0$. Birational morphisms of $A^2$ are mappings $A^2 \to A^2$ given by polynomial mappings $\varphi$ of the polynomial algebra $K[x,y]$ such that for the quotient fields, one has $K(\varphi(x), \varphi(y)) = K(x,y)$. Polynomial automorphisms are obvious examples of such mappings. Another obvious example is the mapping $\tau_x$ given by $x \to x, ~y \to xy$. For a while, it was an open question whether every birational morphism is a product of polynomial automorphisms and copies of $\tau_x$. This question was answered in the negative by P. Russell (in an informal communication). In this paper, we give a simple combinatorial solution of the same problem. More importantly, our method yields an algorithm for deciding whether a given birational morphism can be factored that way.