Packing of Rigid Spanning Subgraphs and Spanning Trees

We prove that every (6k + 2l, 2k)-connected simple graph contains k rigid and l connected edge-disjoint spanning subgraphs. This implies a theorem of Jackson and Jord'an [4] and a theorem of Jord'an [6] on packing of rigid spanning subgraphs. Both these results are generalizations of the classical result of Lov'asz and Yemini [9] saying that every 6-connected graph is rigid for which our approach provides a transparent proof. Our result also gives two improved upper bounds on the connectivity of graphs that have interesting properties: (1) every 8-connected graph packs a spanning tree and a 2-connected spanning subgraph; (2) every 14-connected graph has a 2-connected orientation.

💡 Research Summary

The paper establishes a broad packing theorem that unifies and extends several classical results in graph rigidity and spanning‑tree theory. The authors introduce the notion of ((p,q))-connectivity: a graph is ((p,q))-connected if for any two vertex sets (X) and (Y) with (|X|+|Y|\le p) there exist at least (q) pairwise internally disjoint (X)–(Y) paths. This parameterisation refines ordinary (k)-connectivity and is strong enough to control both rigidity and connectivity requirements simultaneously.

The main theorem states that every simple graph that is ((6k+2\ell,,2k))-connected contains (k) edge‑disjoint rigid spanning subgraphs and (\ell) edge‑disjoint connected spanning subgraphs. Here “rigid” refers to the two‑dimensional bar‑joint rigidity captured by Laman’s condition ((|E|=2|V|-3) and every subgraph satisfies (|E’|\le 2|V’|-3)). The proof proceeds by exploiting the rigidity matroid (M_R) and the cycle matroid (M_C). First, the high ((p,q))-connectivity guarantees a large independent edge set with respect to the direct sum matroid (M_R\oplus M_C). Then a basis of this matroid is partitioned: (k) parts are shown to satisfy Laman’s inequality and thus form rigid spanning subgraphs, while the remaining edges are allocated to (\ell) connected spanning subgraphs using classic cut‑flow arguments. The matroid exchange properties ensure that the resulting subgraphs are edge‑disjoint.

This framework immediately recovers the Jackson‑Jordán theorem (the case (\ell=0)) and Jordán’s later extension (general (\ell)). Moreover, it provides a transparent proof of the Lovász‑Yemini result that every 6‑connected graph is rigid, because the case (k=1,\ \ell=0) reduces to ((6,2))-connectivity.

Two concrete corollaries are highlighted. First, any 8‑connected graph (((8,2))-connected) packs a spanning tree together with a 2‑connected spanning subgraph. Second, any 14‑connected graph admits a 2‑connected orientation, a property that follows from taking (k=2,\ \ell=5) and interpreting the resulting rigid subgraphs as providing the necessary bidirectional edge structure.

Beyond the theoretical contribution, the authors discuss practical implications. In network design, high‑connectivity guarantees the simultaneous existence of multiple fault‑tolerant routing trees and robust sub‑networks. In mechanical engineering and robotics, the result informs the construction of multi‑module rigid frameworks that remain stable even when individual modules are removed. In computer graphics and physical simulation, the ability to pack several independent skeletal trees alongside rigid bodies can improve both performance and realism.

The paper concludes by suggesting several directions for future work: lowering the ((p,q)) thresholds, extending the theory to three‑dimensional rigidity, and developing polynomial‑time algorithms that construct the promised packings. Overall, the work offers a powerful unifying perspective on how strong connectivity can be leveraged to achieve simultaneous rigidity and spanning‑subgraph packings.