In this paper we investigate the Rees algebras of squarefree monomial ideals \(I \subset S=K[x_1,\dots ,x_n]\) generated in degree \(n-2\) , where K is a field. Every such ideal arises as the complementary edge ideal \(I_c(G)\) of a finite simple graph G. We describe the defining equations of the Rees algebra \(\mathcal {R}(I_c(G))\) in terms of the combinatorics of G. If G is a tree or a unicyclic graph whose unique induced cycle has length 3 or 4, we prove that \(\mathcal {R}(I_c(G))\) is Koszul. We also determine the asymptotic depth of the powers of \(I_c(G)\) , proving that \(\lim _{k \rightarrow \infty }\text {depth}\, S/I_c(G)^k=b(G)\) , where b(G) is the number of bipartite connected components of G. Finally, we show that the index of depth stability of \(I_c(G)\) is at most \(n-2\) , and equality holds when G is a path graph.
