  For graphs $G$ and $H$, we say that $G$ is $H$-free if it does not contain
$H$ as an induced subgraph. Already in the early 1980s Alekseev observed that
if $H$ is connected, then the \textsc{Max Weight Independent Set} problem
(MWIS) remains \textsc{NP}-hard in $H$-free graphs, unless $H$ is a path or a
subdivided claw, i.e., a graph obtained from the three-leaf star by subdividing
each edge some number of times (possibly zero). Since then determining the
complexity of MWIS in these remaining cases is one of the most important
problems in algorithmic graph theory.
  A general belief is that the problem is polynomial-time solvable, which is
witnessed by algorithmic results for graphs excluding some small paths or
subdivided claws. A more conclusive evidence was given by the recent
breakthrough result by Gartland and Lokshtanov [FOCS 2020]: They proved that
MWIS can be solved in quasipolynomial time in $H$-free graphs, where $H$ is any
fixed path. If $H$ is an arbitrary subdivided claw, we know much less: The
problem admits a QPTAS and a subexponential-time algorithm [Chudnovsky et al.,
SODA 2019].
  In this paper we make an important step towards solving the problem by
showing that for any subdivided claw $H$, MWIS is polynomial-time solvable in
$H$-free graphs of bounded degree.
