  We give a structural description of the class $\cal C$ of graphs that do not
contain a cycle with a unique chord as an induced subgraph. Our main theorem
states that any connected graph in $\cal C$ is either in some simple basic
class or has a decomposition. Basic classes are chordless cycles, cliques,
bipartite graphs with one side containing only nodes of degree two and induced
subgraphs of the famous Heawood or Petersen graph. Decompositions are node
cutsets consisting of one or two nodes and edge cutsets called 1-joins. Our
decomposition theorem actually gives a complete structure theorem for ${\cal
C}$, i.e. every graph in ${\cal C}$ can be built from basic graphs that can be
explicitly constructed, and gluing them together by prescribed composition
operations; and all graphs built this way are in ${\cal C}$.
  This has several consequences: an ${\cal O}(nm)$-time algorithm to decide
whether a graph is in $\cal C$, an ${\cal O}(n+m)$-time algorithm that finds a
maximum clique of any graph in $\cal C$ and an ${\cal O}(nm)$-time coloring
algorithm for graphs in $\cal C$. We prove that every graph in $\cal C$ is
either 3-colorable or has a coloring with $\omega$ colors where $\omega$ is the
size of a largest clique. The problem of finding a maximum stable set for a
graph in $\cal C$ is known to be NP-hard.
