  DP-coloring is a generalization of list coloring that was introduced in 2015
by Dvo\v{r}\'{a}k and Postle. The chromatic polynomial of a graph $G$, denoted
$P(G,m)$, is equal to the number of proper $m$-colorings of $G$. A well-known
tool for computing the chromatic polynomial of graph $G$ is the
deletion-contraction formula which relates $P(G,m)$ to the chromatic
polynomials of two smaller graphs. The DP color function of a graph $G$,
denoted $P_{DP}(G,m)$, is a DP-coloring analogue of the chromatic polynomial,
and $P_{DP}(G,m)$ is the minimum number of DP-colorings of $G$ over all
possible $m$-fold covers. In this paper we present a deletion-contraction
relation for the DP color function. To make this possible, we extend the
definition of the DP color function to multigraphs. We also introduce the dual
DP color function of a graph $G$, denoted $P^*_{DP}(G,m)$, which counts the
maximum number of DP-colorings of $G$ over certain $m$-fold covers. We show how
the dual DP color function along with our deletion-contraction relation yields
a new general lower bound on the DP color function of a graph.
