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 we are given a graph G = (V,E), a multiset R = {r1, . . . , rt} of vertices in V , and a matroid M on R. We prove a necessary and sufficient condition for G to be decomposed into t edge-disjoint subgraphs G1 = (V1, T1), . . . , Gt = (Vt, Tt) such that (i) for each i, Gi is a tree with ri ∈ Vi, and (ii) for each v ∈ V , the multiset {ri ∈ R | v ∈ Vi} is a base of M. If M is a free matroid, this is a decomposition into t edge-disjoint spanning trees; thus, our result is a proper extension of Nash-Williams’ tree-partition theorem. Such a matroid constraint is motivated by combinatorial rigidity theory. As a direct application of our decomposition theorem, we present characterizations of the infinitesimal rigidity of frameworks with non-generic “boundary”, which extend classical Laman’s theorem for generic 2-rigidity of bar-joint frameworks and Tay’s theorem for generic d-rigidity of body-bar frameworks.