  Considering connected $K_{1,3}$-free graphs with independence number at least
$3$, Chudnovsky and Seymour (2010) showed that every such graph, say $G$, is
$2\omega$-colourable where $\omega$ denotes the clique number of $G$. We study
$(K_{1,3}, Y)$-free graphs, and show that the following three statements are
equivalent.
  (1) Every connected $(K_{1,3}, Y)$-free graph which is distinct from an odd
cycle and which has independence number at least $3$ is perfect.
  (2) Every connected $(K_{1,3}, Y)$-free graph which is distinct from an odd
cycle and which has independence number at least $3$ is $\omega$-colourable.
  (3) $Y$ is isomorphic to an induced subgraph of $P_5$ or $Z_2$ (where $Z_2$
is also known as hammer).
  Furthermore, for connected $(K_{1,3}, Y)$-free graphs (without an assumption
on the independence number), we show a similar characterisation featuring the
graphs $P_4$ and $Z_1$ (where $Z_1$ is also known as paw).
