  Given two graphs $G$ and $H$, the {Ramsey number} $R(G,H)$ is the smallest
positive integer $N$ such that every 2-coloring of the edges of $K_{N}$
contains either a red $G$ or a blue $H$. Let $K_{N-1}\sqcup K_{1,k}$ be the
graph obtained from $K_{N-1}$ by adding a new vertex $v$ connecting $k$
vertices of $K_{N-1}$. Hook and Isaak (2011) defined the {\em star-critical
Ramsey number} $r_{*}(G,H)$ as the smallest integer $k$ such that every
2-coloring of the edges of $K_{N-1}\sqcup K_{1,k}$ contains either a red $G$ or
a blue $H$, where $N=R(G, H)$. For sufficiently large $n$, Li and
Rousseau~(1996) proved that $R(K_{k+1},K_{1}+nK_{t})=knt +1$, Hao, Lin~(2018)
showed that $r_{*}(K_{k+1},K_{1}+nK_{t})=(k-1)tn+t$;
  Li and Liu~(2016) proved that $R(C_{2k+1}, K_{1}+nK_{t})=2nt+1$, and Li, Li,
and Wang~(2020) showed that $r_{*}(C_{2m+1},K_{1}+nK_{t})=nt+t$. A graph $G$
with $\chi(G)=k+1$ is called edge-critical if $G$ contains an edge $e$ such
that $\chi(G-e)=k$. In this paper, we extend the above results by showing that
for an edge-critical graph $G$ with $\chi(G)=k+1$, when $k\geq 2$, $t\geq 2$
and $n$ is sufficiently large, $R(G, K_{1}+nK_{t})=knt+1$ and
$r_{*}(G,K_{1}+nK_{t})=(k-1)nt+t$.
