### On restricted domination in graphs

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For a graph property $\mathcal{P}$ and a graph $G$, we define the domination subdivision number with respect to the property $\mathcal{P}$ to be the minimum number of edges that must be subdivided (where each edge in $G$ can be subdivided at most once) in order to change the domination number with respect to the property $\mathcal{P}$. In this paper we obtain upper bounds in terms of maximum degree and orientable/non-orientable genus for the domination subdivision number with respect to an induced-hereditary property, total domination...

In this paper we present results on changing and unchanging of the domination number with respect to the nondegenerate property $\mathcal{P}$, denoted by ${\gamma}_{\mathcal{P}}\left(G\right)$, when a graph $G$ is modified by deleting a vertex or deleting edges. A graph $G$ is ${({\gamma}_{\mathcal{P}}\left(G\right),k)}_{\mathcal{P}}$-critical if ${\gamma}_{\mathcal{P}}(G-S)<{\gamma}_{\mathcal{P}}\left(G\right)$ for any set $S\u228aV\left(G\right)$ with $\left|S\right|=k$. Properties of ${({\gamma}_{\mathcal{P}},k)}_{\mathcal{P}}$-critical graphs are studied. The plus bondage number with respect to the property $\mathcal{P}$, denoted ${b}_{\mathcal{P}}^{+}\left(G\right)$, is the cardinality of the smallest set of edges $U\subseteq E\left(G\right)$ such that ${\gamma}_{\mathcal{P}}(G-U)>{\gamma}_{\mathcal{P}}\left(G\right)$. Some known results for ordinary domination and bondage numbers...

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 $\mathcal{P}$, is the minimum cardinality of a dominating $\mathcal{P}$-set. In this paper we present results on changing and unchanging of the domination number with respect to the nondegenerate and hereditary properties when a graph is modified by adding an edge or deleting a vertex.

The paper studies minimal acyclic dominating sets, acyclic domination number and upper acyclic domination number in graphs having cut-vertices.

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