Complements of trees

Determine whether the n-vertex complement of an m-vertex tree is ensnaring, proving in particular that for even n greater than m it is ensnaring if and only if m is odd.

Background

The paper develops structural results for n-vertex complements of connected components of complement graphs and completely analyzes complements of complete graphs and complete bipartite graphs in specified parameter ranges. It then identifies complements of trees as a natural unresolved family, proposing a parity-based characterization when the number of added isolated complement vertices makes n even.

References

With an eye toward understanding dense graphs, it is natural to ask when the $n$-vertex complement of a tree is ensnaring.

— Homology in Combinatorial Refraction Billiards  (2502.06013 - Defant et al., 9 Feb 2025) in Conjecture 6.1, Section 6, Future Directions, subsection Ensnaring Graphs