Block-and-hole graphs: Constructibility and $(3,0)$-sparsity

TL;DR

This paper characterizes minimally 3-rigid block-and-hole graphs with one block or hole using constructibility from K3 and $(3,0)$-tight face graphs, linking graph theory, rigidity, and polyhedral origami.

Contribution

It provides a new characterization of minimally 3-rigid block-and-hole graphs via constructibility and $(3,0)$-tightness, with polynomial-time verification methods.

Findings

Characterization of minimally 3-rigid block-and-hole graphs.

Polynomial-time pebble game algorithm for $(3,0)$-tightness.

Connections to origami and $oldsymbol{ eq}$-2 rigidity in $oldsymbol{ eq}$-space.

Abstract

We show that minimally 3-rigid block-and-hole graphs, with one block or one hole, are characterised as those which are constructible from $K_3$ by vertex splitting, and also, as those having associated looped face graphs which are $(3,0)$-tight. This latter property can be verified in polynomial time by a form of pebble game algorithm. We also indicate connections to the rigidity properties of polyhedral surfaces known as origami and to graph rigidity in $\ell_p^3$ for $p\not=2$.