Rigidity of nearly planar classes of graphs

Abstract

We explore the rigidity of generic frameworks in 3-dimensions whose underlying graph is close to being planar. Specifically we consider apex graphs, edge-apex graphs and their variants and prove independence results in the generic 3-dimensional rigidity matroid adding to the short list of graph classes for which 3-dimensional rigidity is understood. We then analyse global rigidity for these graph classes and use our results to deduce bounds on the maximum likelihood threshold of graphs in these nearly planar classes.

Figures (9)

• Figure 1: Examples of apex and edge-apex graphs.
• Figure 2: $K_5$ on the projective plane and $K_5 \overline{(a,e)} K_5$ on the Klein Bottle.
• Figure 3: Torus map of $K_5 \overline{(a,e)} K_5$.
• Figure 4: The double banana $D\!B$.
• Figure 5: A triangulation and its decomposition into 4-blocks.

Theorems & Definitions (58)

• Theorem 1.1: MR0400239
• Lemma 2.1
• proof
• Theorem 2.2
• Lemma 2.3: Maxwell
• Lemma 2.4: Wlong
• Lemma 2.5: Wlong
• Lemma 2.6: ConingGGJ
• Theorem 3.1: MR927685
• Theorem 3.2