Rooted-Tree Decompositions with Matroid Constraints and the Infinitesimal Rigidity of Frameworks with Boundaries

Source abstract

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 prove a necessary and sufficient condition for $G$ to be decomposed into $t$ edge-disjoint subgraphs $G_1=(V_1,T_1), \dots, G_t=(V_t,T_t)$ such that (i) for each $i$, $G_i$ is a tree with $r_i\in V_i$, and (ii) for each $v\in V$, the multiset $\{r_i\in {\bm R} \mid v\in V_i\}$ is a base of ${\cal M}$. If ${\cal M}$ 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 nongeneric “boundary,” which extend classical the Laman's theorem for generic 2-rigidity of bar-joint frameworks and Tay's theorem for generic $d$-rigidity of body-bar frameworks.