On kP3-free graphs, the maximum weight list r-colorable induced subgraph problem is solvable in polynomial time for every fixed r and k, as are the distance-d versions for d at least 6.
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Maximum list $r$-colorable induced subgraphs in $kP_3$-free graphs
On kP3-free graphs, the maximum weight list r-colorable induced subgraph problem is solvable in polynomial time for every fixed r and k, as are the distance-d versions for d at least 6.