  A decomposition of a graph $G$ is a family of subgraphs of $G$ whose edge
sets form a partition of $E(G)$. In this paper, we prove that every
triangle-free planar graph $G$ can be decomposed into a $2$-degenerate graph
and a matching. Consequently, every triangle-free planar graph $G$ has a
matching $M$ such that $G-M$ is online 3-DP-colorable. This strengthens an
earlier result in [R. \v{S}krekovski, {\em A Gr\"{o}tzsch-Type Theorem for List
Colourings with Impropriety One}, Combin. Prob. Comput. 8 (1999), 493-507] that
every triangle-free planar graph is $1$-defective $3$-choosable.
