List-constrained point-set embeddability is poly-time or FPT for connected graphs on convex points, but NP-hard already for bi-labeled matchings (convex) and bi-labeled paths (general), with matching FPT/paraNP dichotomies by vertex cover and list size.
On embedding an outer-planar graph in a point set , year =
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Point Set Embeddability with List Constraints
List-constrained point-set embeddability is poly-time or FPT for connected graphs on convex points, but NP-hard already for bi-labeled matchings (convex) and bi-labeled paths (general), with matching FPT/paraNP dichotomies by vertex cover and list size.