  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. Chromatic
polynomials for joins and vertex-gluings of graphs are well understood, but the
effect of these graph operations on the DP color function is not known. In this
paper we make progress on understanding the DP color function of the join of a
graph with a complete graph and vertex-gluings of certain graphs. We also
develop tools to study the DP color function under these graph operations, and
we study the threshold (smallest $m$) beyond which the DP color function of a
graph constructed with these operations equals its chromatic polynomial.
