AI 中文总结
研究连通带状图拟树计数问题，证明其在多项式时间图灵归约下是#P完全的，还通过带框Cohn - Lempel等式等将计数与交错多项式求值联系起来，同时给出可定向带状图等情形的行列式证明。
AI 中文摘要
连通带状图的拟树是具有恰好一个边界分量的生成带状子图；拟树在嵌入图的拓扑图论中扮演生成树的角色。我们证明，在多项式时间图灵归约下，即使对于花束图，对其计数也是#P完全的。证明通过自然标识将每个非空带框弦图与配备有特殊A - 迹的4 - 正则映射对应起来，使得拟树对应于A - 迹，根据Ge和Štefankovič的一个定理，对A - 迹计数是#P完全的。通过带框Cohn - Lempel等式，计数也是交错多项式求值 - $q(H;2,1)$，即图的带环圆图$H$的满秩诱导子图的数量，这使其处于Bläser和Hoffmann复杂性分类中未解决的$y = 1$这条线上；然后一个克隆论证使得这条线上除平凡点$(1,1)$外的每个固定有理点在带环圆图上都是#P难的，即使提供了带框弦表示。在可处理方面，相同的GF(2)模型给出了已知行列式情形的简短证明：对于可定向带状图，计数是一个行列式，本质上是Merino、Moffatt和Noble的矩阵 - 拟树定理，在此通过Bouchet的主幺模性证明；对于恰好有一个不可定向环的花束图，它是两个可定向行列式的和，通过一个一阶行列式恒等式等同于Deng、Jin和Yan的行列式公式。
英文摘要
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.
CommentsConcluding section expanded (torsor structures, after Baker-Ding-Kim); main theorems unchanged
