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Rooted-Tree Decompositions with Matroid Constraints and the Infinitesimal Rigidity of Frameworks with Boundaries
As an extension of a classical tree-partition problem, we consider decompositions of graphs into edge-disjoint (rooted-)trees with an additional matroid constraint. Specifically, suppose that we are given a graph $G=(V,E)$, a multiset ${\bm R} = \{r_1,\dots, r_t\}$ of vertices in $V$, and a matroid ${\cal M}$ on ${\bm R}$. We …