SHARP COLLAPSIBILITY BOUNDS FOR NERVES OF SEPARATED $\boldsymbol{d}$-TREES AND PRODUCTS OF TREES

Source abstract

Subtrees of finite trees form one of the basic graph-theoretic intersection models. Motivated by recent collapsibility results for separated $d$-intervals and for axis-parallel boxes, we study nerves of two tree-based intersection models. In the separated model, a set is a disjoint union of subtrees in prescribed tree components. In the product model, a set is a Cartesian product of subtrees of finite trees. We prove that the nerve of every finite family of separated $d$-trees is $(2 d-1)$-collapsible, whereas the nerve of every finite family of products of $d$ subtrees is $d$-collapsible. For finite traces, we obtain a uniform $(D-1)$-collapsibility bound in both models, where $D$ is the total number of leaves of the underlying trees. The bounds are sharp. > These results extend collapsibility phenomena from path-like interval and box models to graph-theoretic tree models, and show that branching, measured by the number of leaves, controls the finite-trace case.