Forbidding the subdivided claw as a subgraph or a minor

Let $Y$ be the subdivided claw, the $7$-vertex tree obtained from a claw $K_{1,3}$ by subdividing each edge exactly once. We characterize the graphs (finite and infinite) that do not have $Y$ as a subgraph, or, equivalently, do not have $Y$ as a minor. This work was motivated by a problem involving VCD minors. A graph $H$ is a vertex contraction-deletion minor, or VCD minor, of a graph $G$ if $H$ can be obtained from $G$ by a sequence of vertex deletions or contractions of all edges incident with a single vertex. Our result is a key step in describing $K_{1,3}$-VCD-minor-free line graphs. We also characterize graphs that forbid each subtree of $Y$. We discuss the relevance of our results for Turán. numbers of trees, and pathwidth and growth constants for graphs without a particular tree as a minor.