The minimum semidefinite rank (msr) of a graph is the minimum rank among positive semidefinite matrices with the given graph. The OS-number is a useful lower bound for msr, which arises by considering ordered vertex sets with some connectivity properties. In this paper, we develop two new interpretations of the OS-number. We first show that OS-number is also equal to the maximum number of vertices which can be orthogonally removed from a graph under certain nondegeneracy conditions. Our second interpretation of the OS-number is as the maximum possible rank of chordal supergraphs who exhibit a notion of connectivity we call isolation-preserving. These interpretations not only give insight into the OS-number, but also allow us to prove some new results. For example we show that msr(G) = G - 2 if and only if OS(G) = Gzsfnc - 2. © 2011 Elsevier Inc. All rights reserved.
Mitchell, L. H., Narayan, S. K., & Zimmer, A. M. (2012). Lower bounds for minimum semidefinite rank from orthogonal removal and chordal supergraphs. Linear Algebra and Its Applications, 436(3), 525–536. https://doi.org/10.1016/j.laa.2011.07.004