Thin Trees via $k$-Respecting Cut Identities

Thin spanning trees lie at the intersection of graph theory, approximation algorithms, and combinatorial optimization. They are central to the long-standing \emph{thin tree conjecture}, which asks whether every $k$-edge-connected graph contains an $O(1/k)$-thin tree, and they underpin algorithmic breakthroughs such as the $O(\log n/\log\log n)$-approximation for ATSP. Yet even the basic algorithmic task of \emph{verifying} that a given tree is thin has remained elusive: checking thinness requires reasoning about exponentially many cuts, and no efficient certificates have been known. We introduce a new machinery of \emph{$k$-respecting cut identities}, which express the weight of every cut that crosses a spanning tree in at most $k$ edges as a simple function of pairwise ($2$-respecting) cuts. This yields a tree-local oracle that, after $O(n^2)$ preprocessing, evaluates such cuts in $O_k(1)$ time. Building on this oracle, we give the first procedure to compute the exact $k$-thinness certificate $\Theta_k(T)$ of any spanning tree for fixed $k$ in time $\tilde O(n^2+n^k)$, outputting both the certificate value and a witnessing cut. Beyond general graphs, our framework yields sharper guarantees in structured settings. In planar graphs, duality with cycles and dual girth imply that every spanning tree admits a verifiable certificate $\Theta_k(T)\le k/\lambda$ (hence $O(1/\lambda)$ for constant $k$). In graphs embedded on a surface of genus $\gamma$, refined counting gives certified (per-cut) bounds $O((\log n+\gamma)/\lambda)$ via the same ensemble coverage.