  For a graphical property $\mathcal{P}$ and a graph $G$, a subset $S$ of
vertices of $G$ is a $\mathcal{P}$-set if the subgraph induced by $S$ has the
property $\mathcal{P}$. The domination number with respect to the property
$\mathcal{P}$, denoted by $\gamma_{\mathcal{P}} (G)$, is the minimum
cardinality of a dominating $\mathcal{P}$-set. We define the domination
multisubdivision number with respect to $\mathcal{P}$,denoted by
$msd_{\mathcal{P}}(G)$, as a minimum positive integer $k$ such that there
exists an edge which must be subdivided $k$ times to change $\gamma_\mathcal{P}
(G)$. In this paper (a) we present necessary and sufficient conditions for a
change of $\gamma_{\mathcal{P}}(G)$ after subdividing an edge of $G$ once, (b)
we prove that if $e$ is an edge of a graph $G$ then $\gamma_\mathcal{P}
(G_{e,1}) < \gamma_\mathcal{P} (G)$ if and only if $\gamma_\mathcal{P} (G-e) <
\gamma_\mathcal{P} (G)$ ($G_{e,t}$ denote the graph obtained from $G$ by
subdivision of $e$ with $t$ vertices), (c) we also prove that for every edge of
a graph $G$ is fulfilled $\gamma_{\mathcal{P}}(G-e) \leq
\gamma_{\mathcal{P}}(G_{e,3}) \leq \gamma_{\mathcal{P}}(G-e) + 1$, and (d) we
show that $msd_{\mathcal{P}}(G) \leq 3$, where $\mathcal{P}$ is hereditary and
closed under union with $K_1$.
