Computational complexity of clique and independence numbers

Determine whether, for every fixed t ≥ 2, computing the clique number or independence number of the generalized spanning tree t-disjointness graph Γ_t(G) is NP-hard when G is an arbitrary graph.

Background

For an arbitrary graph G, Γ_t(G) has the spanning trees of G as vertices, with two trees adjacent when they have fewer than t common edges. The authors ask whether the associated clique and independence number problems are computationally intractable for fixed t, extending the extremal-combinatorial questions from complete graphs to arbitrary graphs.