Globally linked pairs and cheapest globally rigid supergraphs

TL;DR

Every rigid graph in $\mathbb{R}^2$ has a tree-like structure, which conveys all the information regarding its globally rigid augmentations, and a new, simple solution to the minimum cardinality version for rigid input graphs, a problem which is known to be solvable in polynomial time.

Abstract

Given a graph $G$, a cost function on the non-edges of $G$, and an integer $d$, the problem of finding a cheapest globally rigid supergraph of $G$ in $\mathbb{R}^d$ is NP-hard for $d\geq 1$. For this problem, which is a common generalization of several well-studied graph augmentation problems, no approximation algorithm has previously been known for $d\geq 2$. Our main algorithmic result is a 5-approximation algorithm in the $d=2$ case. We achieve this by proving numerous new structural results on rigid graphs and globally linked vertex pairs. In particular, we show that every rigid graph in $\mathbb{R}^2$ has a tree-like structure, which conveys all the information regarding its globally rigid augmentations. Our results also yield a new, simple solution to the minimum cardinality version (where the cost function is uniform) for rigid input graphs, a problem which is known to be solvable in polynomial time.

Figures (5)

• Figure 1: Two pairs of equivalent generic frameworks of a graph $G$ in $\mathbb{R}^2$. The vertex pair $\{u,v\}$ is globally linked in the two frameworks on the left. On the other hand, $\{u,v\}$ is not globally linked in the two frameworks on the right. Thus $\{u,v\}$ is weakly globally linked, but not globally linked in $G$ in $\mathbb{R}^2$.
• Figure 2: Lemma \ref{['coro']} gives a sufficient condition for the weak global linkedness of $\{u,v\}$.
• Figure 3: The two graphs on the left are rigid and $2$-special. The third graph is SNGR in $\mathbb{R}^2$, while the rightmost graph is totally loose in $\mathbb{R}^2$.
• Figure 4: The graph $G$ on the left is a rigid totally loose graph. On the right the minimal non-complete standard subgraphs of $G$ are depicted. (Overall $G$ has 24 non-complete standard subgraphs. Five of these are depicted here. The remaining 19 can be obtained by taking the union of some of these five subgraphs such that the resulting graph is connected.)
• Figure 5: A rigid totally loose graph and its tree representation. Each vertex of the tree representation represents a subgraph of $G$, which is depicted next to the vertex. The three different types of vertices in the tree representation are indicated by the different shapes. (square - 3-connected SNGR graph; circle - clique; diamond - 2-separator)

Theorems & Definitions (77)

• Theorem 2.1
• Theorem 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• Lemma 3.6
• Lemma 3.7