Euler and face data from a planar drawing

Given the stated crossing-free drawing on at least three vertices, extend it to a connected planar supergraph K on the same vertices and obtain its Euler face count f and total facial boundary length l. The outputs assert that K contains H, its edge count dominates H's, Euler's equation holds, each edge contributes two face sides, and every facial boundary has length at least three in total. A proof route is to regard the finite family of simple arcs as a plane embedding, add noncrossing edges until the embedding is maximal on the fixed vertex set, and count the facial boundary walks; maximality gives triangular faces, including the three-vertex boundary case. This is the substantive remaining topological obligation; the parent uses only the edge-count monotonicity and numerical data.