Classes of graphs with minimum skew rank 4

The minimum skew rank of a simple graph G   is the smallest possible rank among all real skew-symmetric matrices whose (i,j)$(i,j)$-entry is nonzero if and only if the edge joining i and j is in G. It is known that a graph has minimum skew rank 2 if and only if it consists of a complete multipartite graph and some isolated vertices. Some necessary conditions for a graph to have minimum skew rank 4 are established, and several families of graphs with minimum skew rank 4 are given. Linear algebraic techniques are developed to show that complements of trees and certain outerplanar graphs have minimum skew rank 4.