Adnan Sljoka KGU/CREST, Japan How to inductively construct floating body-hinge / molecular Assur graphs?

Def’n. A graph G = (V,E) is (6, 6)-sparse if for every subgraph G′ = (V′,E′), |E′| ≤ 6|V ′| − 6. G is (6, 6)-tight if G is (6, 6)-sparse and |E| = 6|V |− 6. Theorem. (Tay-Whiteley): A 3D (generic) body-hinge framework on a graph H = (V, E) (vertices correspond to bodies, edges to hinges) is minimally rigid if and only if 5H is (6,6)-tight. 5H denotes a multigraph obtained from H where each edge e (hinge) in H is replaced by 5 parallel copies of e. Two bodies joined with a hinge H 5H Not minimally rigid. Minimally rigid.

Problem: How to inductively construct all graphs H such that 5H is (6,6)-tight with no proper (6,6)-tight subgraph (1<|V′|<|V|)? (i.e. minimally rigid 3D body-hinge graph with no proper rigid subgraph) These graphs occur in applications in mechanical engineering and robotics – Assur graphs. 3D Floating body-hinge Assur graphs – Minimally rigid body-hinge graph with no proper rigid subgraph. For any vertex (body) we pin, removing any single edge in 5H induces motion of all inner vertices. Not BH Assur graphNot Floating BH Assur graphs Inductive construction of Assur graphs useful to Mechanical Engineers for synthesizing new mechanisms.

Starting with a pinned minimally rigid body-hinge graph H in 3D we can always generate a 6-directed orientation of 5H (i.e. with pebble game), where every inner vertex has out-degree 6 and the pinned vertex out-degree 0. In floating body-hinge Assur graphs, for any vertex (body) we pin, the set of inner vertices will be strongly connected in the directed graph.

Thank you.

Body-hinge Assur graph decomposition. Given a minimal rigid body-hinge graph H and specified pin. Start with 5H and generate directions (via pebble game) towards the ground. Every pinned minimally rigid body-hinge graph in 3D has a 6-directed orientation, where every inner vertex has out-degree 6 and pinned vertex out-degree 0.

Find strongly connected components of directed graph Each strongly connected components with its outgoing edges becoming pinned to the ground is a Body-Hinge Assur graph with respect to specified pin (no necessarily floating Body-Hinge Assur graph)

Note in the floating Body-Hinge Assur graph, for any pin all inner vertices will be strongly connected. Floating Assur graphs Not Floating Assur graphs Some components may not be floating Assur graphs A B C D D C A

D Not floating body-hinge Assur graph, so we may get more than one strongly connected component for some pin Floating body-hinge Assur graph, for any pin, all vertices are strongly connected.