gr.group theory - "Remove a vertex" map for right-angled Artin groups

No. Let $Gamma$ be the graph with two vertices and no edges - the non-abelian free group of rank two - and let $g$ be the commutator of the two generators $s_1$ and $s_2$. Then $g$ is certainly non-trivial, but $g$ dies whenever you kill $s_1$ or $s_2$.

UPDATE:

For an example with a connected graph, let's take $Gamma$ to be the straight-line graph with four vertices $a,b,c,d$ (so $[a,b]=[b,c]=[c,d]=1$). Now consider $g=[[c,a],[b,d]]$. Clearly this dies when you kill any generator. On the other hand,

$g=cac^{-1}a^{-1}bdb^{-1}d^{-1}aca^{-1}c^{-1}dbd^{-1}b^{-1}$

and a well-known solution to the word problem in right-angled Artin groups tells you that $g$ is non-trivial.