Spanning rigid subgraph packing and sparse subgraph covering

TL;DR

This paper establishes new sufficient conditions for packing spanning rigid subgraphs and spanning trees in highly connected graphs, and explores sparse subgraph covering, advancing understanding in graph rigidity and connectivity.

Contribution

It introduces a novel partition condition for packing spanning rigid subgraphs and trees, improving bounds on connectivity requirements for such packings.

Findings

A graph with $(4k+2l)$-edge connectivity contains a packing of $k$ spanning rigid subgraphs and $l$ spanning trees.

Every 6-connected and essentially 8-connected graph has a spanning tree whose removal leaves a 2-connected graph.

The results improve previous bounds on connectivity for rigid subgraph packing and sparse covering.

Abstract

Rigidity, arising in discrete geometry, is the property of a structure that does not flex. Laman provides a combinatorial characterization of rigid graphs in the Euclidean plane, and thus rigid graphs in the Euclidean plane have applications in graph theory. We discover a sufficient partition condition of packing spanning rigid subgraphs and spanning trees. As a corollary, we show that a simple graph $G$ contains a packing of $k$ spanning rigid subgraphs and $l$ spanning trees if $G$ is $(4k+2l)$-edge-connected, and $G-Z$ is essentially $(6k+2l - 2k|Z|)$-edge-connected for every $Z\subset V(G)$. Thus every $(4k+2l)$-connected and essentially $(6k+2l)$-connected graph $G$ contains a packing of $k$ spanning rigid subgraphs and $l$ spanning trees. Utilizing this, we show that every $6$-connected and essentially $8$-connected graph $G$ contains a spanning tree $T$ such that $G-E(T)$ is $2$-connected. These improve some previous results. Sparse subgraph covering problems are also studied.