  We prove a decomposition theorem for graphs that do not contain a subdivision
of $K_4$ as an induced subgraph where $K_4$ is the complete graph on four
vertices. We obtain also a structure theorem for the class $\cal C$ of graphs
that contain neither a subdivision of $K_4$ nor a wheel as an induced subgraph,
where a wheel is a cycle on at least four vertices together with a vertex that
has at least three neighbors on the cycle. Our structure theorem is used to
prove that every graph in $\cal C$ is 3-colorable and entails a polynomial-time
recognition algorithm for membership in $\cal C$. As an intermediate result, we
prove a structure theorem for the graphs whose cycles are all chordless.
