  DP-coloring (also called correspondence coloring) is a generalization of list
coloring that has been widely studied in recent years after its introduction by
Dvo\v{r}\'{a}k and Postle in 2015. As the analogue of the chromatic polynomial
$P(G,m)$, the DP color function of a graph $G$, denoted $P_{DP}(G,m)$, counts
the minimum number of DP-colorings over all possible $m$-fold covers. It is
known that, unlike the list color function $P_{\ell}(G,m)$, for any $g \geq 3$
there exists a graph $G$ with girth $g$ such that $P_{DP}(G,m) < P(G,m)$ when
$m$ is sufficiently large. Thus, two fundamental open questions regarding the
DP color function are: (i) for which $G$ does there exist an $N \in \mathbb{N}$
such that $P_{DP}(G,m) = P(G,m)$ whenever $m \geq N$, (ii) Given a graph $G$
does there always exist an $N \in \mathbb{N}$ and a polynomial $p(m)$ such that
$P_{DP}(G,m) = p(m)$ whenever $m \geq N$?
  In this paper we give exact formulas for the DP color function of a Theta
graph based on the parity of its path lengths. This gives an explicit answer,
including the formulas for the polynomials that are not the chromatic
polynomial, to both the questions above for Theta graphs. We extend this result
to Generalized Theta graphs by characterizing the exact parity condition that
ensures the DP color function eventually equals the chromatic polynomial. To
answer the second question for Generalized Theta graphs, we confirm it for the
larger class of graphs with a feedback vertex set of size one.
