2026/07/21 by William Whistler
#math.CO #cs.CC
A quasi-tree of a connected ribbon graph is a spanning ribbon subgraph with exactly one boundary component; quasi-trees play the role of spanning trees in the topological graph theory of embedded graphs. We prove that counting them is #P-complete under polynomial-time Turing reductions, already for bouquets. The proof identifies every nonempty framed chord diagram, up to natural identifications, with a 4-regular map equipped with a distinguished A-trail, in such a way that quasi-trees correspond to A-trails, whose counting is #P-complete by a theorem of Ge and Štefankovič. Through the framed Cohn-Lempel equality the count is also an interlace-polynomial evaluation - q(H;2,1), the number of full-rank induced subgraphs of the looped circle graph H of the diagram - placing it on the line y=1 left open in the complexity classification of Bläser and Hoffmann; a cloning argument then makes every fixed rational point of that line, other than the trivial (1,1), #P-hard on looped circle graphs, even when a framed chord representation is supplied. On the tractable side, the same GF(2) model yields short proofs of the known determinantal cases: for orientable ribbon graphs the count is a determinant, essentially the Matrix-Quasi-tree Theorem of Merino, Moffatt and Noble, proved here via Bouchet's principal unimodularity, and for bouquets with exactly one non-orientable loop it is a sum of two orientable determinants, equivalent by a rank-one determinant identity to the determinant formula of Deng, Jin and Yan.