$\text{HAMPATH} = \{(G=(V,E),s',t')| \text{ G has a Hamilton path from s' to t' } \}$
$\text{LONG-PATH} = \{(G,s,t,k) | \text{G has a simple path p from s to t, length(p) $\geq$ k} \}$
I'm trying to prove a polynomial reduction, $\text{HAMPATH} \leq_p \text{LONG-PATH}$.
My thought was to take an instance of $\text{HAMPATH}$ and add exactly $k-|V|$ nodes and edges from $t'$ to $s'$, so i have an Hamilton path from $s=s'$ to $t'$, and a simple path(which is also an Hamilton path) from $t'$ to $t$.
I tried to prove that but i couldn't finish the proof and it seems that this is not good reduction.
any thoughts?
