The 2-dimensional global rigidity has been shown to be equivalent to 3-connectedness and redundant rigidity by a combination of two results due to Jackson and Jordán, and Connelly, respectively. By the characterization, a theorem of Lovász and Yemini implies that every 6-connected graph is redundantly rigid and thus globally rigid. The 6-connectedness is best possible, since there exist infinitely many 5-connected nonrigid graphs. Jackson, Servatius, and Servatius used the idea of “essential connectivity” and proved that every 4-connected “essentially 6-connected” graph is redundantly rigid and thus global rigid. Since 3-connectedness is a necessary condition of global rigidity, it is interesting to study 3-connected graphs for redundant rigidity and thus global rigidity. We utilize a different “essential connectivity” and prove that every 3-connected essentially 9-connected graph is redundantly rigid and thus globally rigid. The essential 9-connectedness is best possible. Under this essential connectivity, we also prove that every 4-connected essentially 6-connected graph is redundantly rigid and thus globally rigid. Our proofs are based on discharging arguments.

For a given graph $H$, a graph $G$ is H-linked if, for every injection $\varphi: V(H) \to V(G)$, the graph $G$ contains a subdivision of $H$ with $\varphi(v)$ corresponding to $v$ for each $v\in V(H)$. Let $f(H)$ be the minimum integer $k$ such that every $k$-connected graph is $H$-linked. Among connected simple graphs $H$ with at least four vertices, the exact value $f(H)$ is only known when $H$ is a star, or a path with four vertices, or a cycle with four vertices. A kite is the graph obtained from $K_4$ by deleting two adjacent edges, i.e., a triangle together with a pendant edge. The exact value of $f(H)$ when $H$ is the kite remains open. In this paper, we settle this problem by showing that every 7-connected graph is kite-linked.