  Let G be a finite simple graph and let indm(G) and ordm(G) denote the induced
matching number and the ordered matching number of G, respectively. We
characterize all bipartite graphs G with indm(G) = ordm(G). We establish the
Castelnuovo-Mumford regularity of powers of edge ideals and depth of powers of
cover ideals for such graphs. We also give formulas for the count of connected
non-isomorphic spanning subgraphs of the complete bipartite graph K_{m,n} for
which indm(G) = ordm(G) = 2, with an explicit expression for the count when m =
2,3,4 and m <= n.
