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For any graphs $G$ and $H$, we say that a bound is of Vizing-type if$\gamma(G\square H)\geq c \gamma(G)\gamma(H)$ for some constant $c$. We showseveral bounds of Vizing-type for graphs $G$ with forbidden induced subgraphs.In particular, if $G$ is a triangle and $K_{1,r}$-free graph, then for anygraph $H$, $\gamma(G\square H)\geq \frac{r}{2r-1}\gamma(G)\gamma(H)$. If $G$ isa $K_r$ and $P_5$-free graph for some integer $r\geq 2$, then for any graph$H$, $\gamma(G\square H)\geq \frac{r-1}{2r-3}\gamma(G)\gamma(H)$. We do this bybounding the power of $G$, $\pi(G)$. We show that if $G$ is claw-free and$P_6$-free or $K_4$ and $P_5$-free, then for any graph $H$, $\gamma(G\squareH)\geq \gamma(G)\gamma(H)$. Furthermore, we show Vizing-type bounds in terms ofthe diameter of $G$.

Vizing-type bounds for graphs with induced subgraph restrictions