List monopolar partitions of claw-free graphs

The list monopolar partition problem asks whether a graph GG together with lists L(v)⊆{0,1},v∈V(G)L(v)⊆{0,1},v∈V(G) admits a mapping f:V(G)→{0,1}f:V(G)→{0,1} such that f(v)∈L(v)f(v)∈L(v) for each v∈V(G),f−1(0) induces an independent set and f−1(1)f−1(1) induces a disjoint union of cliques in GG. This problem is NP-complete in general. We show that the problem is solvable in time O(n2m)O(n2m) for claw-free graphs where nn and mm are the numbers of vertices and edges respectively in the input graph.