  The edit distance between two graphs on the same vertex set is defined to be
the size of the symmetric difference of their edge sets. The edit distance
function of a hereditary property, $\mathcal{H}$, is a function of $p$, and
measures, asymptotically, the furthest graph of edge density $p$ from
$\mathcal{H}$ under this metric. In this paper, we address the hereditary
property $\forb(K_{2,t})$, the property of having no induced copy of the
complete bipartite graph with 2 vertices in one class and $t$ in the other.
Employing an assortment of techniques and colored regularity graph
constructions, we are able to determine the edit distance function over the
entire domain $p\in [0,1]$ when $t=3,4$ and extend the interval over which the
edit distance function for $\forb(K_{2,t})$ is known for all values of $t$,
determining its maximum value for all odd $t$. We also prove that the function
for odd $t$ has a nontrivial interval on which it achieves its maximum. These
are the only known principal hereditary properties for which this occurs.
  In the process of studying this class of functions, we encounter some
surprising connections to extremal graph theory problems, such as strongly
regular graphs and the problem of Zarankiewicz.
  This is an extended version of a paper with the same name now published in
the Journal of Graph Theory \cite{jgt_version}. In particular, this version
contains Appendix A, which has tables and graphs pertaining to the hereditary
property $\forb(K_{2,t})$ for small $t$, and Appendix B, which has the proofs
of Lemma 31, Proposition 32, Proposition 33, and Lemma 34.
