  We algorithmize the recent structural characterization for claw-free graphs
by Chudnovsky and Seymour. Building on this result, we show that Dominating Set
on claw-free graphs is (i) fixed-parameter tractable and (ii) even possesses a
polynomial kernel. To complement these results, we establish that Dominating
Set is not fixed-parameter tractable on the slightly larger class of graphs
that exclude K_{1,4} as an induced subgraph (K_{1,4}-free graphs). We show that
our algorithmization can also be used to show that the related Connected
Dominating Set problem is fixed-parameter tractable on claw-free graphs. To
complement that result, we show that Connected Dominating Set has no polynomial
kernel on claw-free graphs and is not fixed-parameter tractable on K_{1,4}-free
graphs. Combined, our results provide a dichotomy for Dominating Set and
Connected Dominating Set on K_{1,L}-free graphs and show that the problem is
fixed-parameter tractable if and only if L <= 3.
