  Given graphs $G, H_1, H_2$, we write $G \rightarrow ({H}_1, H_2)$ if every
$\{$red, blue$\}$-coloring of the edges of $G$ contains a red copy of $H_1$ or
a blue copy of $H_2$. A non-complete graph $G$ is $(H_1, H_2)$-co-critical if
$G \nrightarrow ({H}_1, H_2)$, but $G+e\rightarrow ({H}_1, H_2)$ for every edge
$e$ in $\overline{G}$. Motivated by a conjecture of Hanson and Toft from 1987,
we study the minimum number of edges over all $(K_3, K_{1,k})$-co-critical
graphs on $n$ vertices. We prove that for all $t\ge3$ and $k\ge 3$, there
exists a constant $\ell(t, k)$ such that, for all $n \ge (t-1)k+1$, if $G$ is a
$(K_t, K_{1,k})$-co-critical graph on $n$ vertices, then $$ e(G)\ge
\left(2t-4+\frac{k-1}{2}\right)n-\ell(t, k).$$ Furthermore, this linear bound
is asymptotically best possible when $t\in\{3, 4,5\}$ and all $k\ge3$ and $n\ge
(2t-2)k+1$. It seems non-trivial to construct extremal $(K_t,
K_{1,k})$-co-critical graphs for $t\ge6$. We also obtain the sharp bound for
the size of $(K_3, K_{1,3})$-co-critical graphs on $n\ge13$ vertices by showing
that all such graphs have at least $3n-4$ edges.
