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This article studies the notions of minimal and 1-redundant bearing rigidity. A necessary and sufficient condition for the numbers of edges in a graph of n(n ≥ 3) vertices to be minimally bearing rigid (MBR) in Rd(d ≥ 2) is proposed. If 3 ≤ n ≤ d + 1, a graph is MBR if and only if it is the cycle graph. In case n > d + 1, a generically bearing rigid graph is minimal if it has precisely 1 + ⌊n−2/d−1⌋ × d + mod(n−2, d−1) + sgn(mod(n−2, d−1)) edges. Then, several conditions for 1-redundant bearing rigidity are derived. Based on the mathematical conditions, some algorithms for generating generically, minimally, and 1-redundantly bearing rigid graphs are given. Furthermore, two applications of the new notions to optimal network design and formation merging are also reported. [ABSTRACT FROM AUTHOR]