Lower Bounds for Minimum Semidefinite Rank from Orthogonal Removal and Chordal Supergraphs
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.
Mitchell, L., Narayan, S. & Zimmer, A. (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.