Definition
(Decorated tropical type.) It’s the same as the notions of type but with the extra decoration of a curve class: we add a map \(A:V(G)\to H^+_2(X)\) mapping components of \(C\) i.e. vertices in \(G\) to curve classes, and we denote the total class \(|A| = \sum_{v\in V(G)} A(v)\). The decorated type is denoted
\begin{align*} \tilde \tau = (\tau, A). \end{align*}This is the level at which the decomposition formula functions; two rigid tropical maps with the same underlying types but with different curve classes are counted separately.