Parity dependence for complements

Prove that for every fixed graph with m vertices, whether its n-vertex complement is ensnaring depends only on the parity of n for all integers n greater than m.

Background

For an m-vertex graph H and n greater than or equal to m, the n-vertex complement is the graph whose complement is the disjoint union of H and n-m isolated vertices. The paper shows that ensnaring behavior can depend on n despite the added vertices having uniform behavior. The conjecture predicts that, once n exceeds the size of the fixed graph, only the parity of n matters.