I am interested in the following variant of the usual Isotopy Extension Theorem:

$\textbf{Question}:$Let $K$ be a graph (1-complex) embedded on a surface $S$, and $i$ be an isotopy of $K$. Does $i$ extend to an isotopy of $S$ (generally called an ambient isotopy or a diffeotopy) ? If $i$ fixes a set of points $V\subseteq K$, does it extend to an ambient isotopy also fixing $V$ ?

$\textbf{Background}$: If $i$ is smooth and $K$ is a submanifold, it is the Isotopy Extension theorem one can find in the literature (for example in Hirsch), which is proved by integrating a vector field. I think that the same argument can be applied in the case of a 1-complex instead of a submanifold since one can still define a tubular neighborhood in this case, but it is a bit far from my field so I am not sure.

I don't know how to get rid of the "smooth" hypothesis. On surfaces, classical results by Munkres (every homeomorphism is homotopic to a diffeomorphism) and Boldsen (or probably others before) (homotopic diffeomorphisms are smoothly isotopic) allow us to suppose everything about ambient isotopies is smooth. But here I start with a non ambient one and try to extend it, so I don't see how to apply them.

Any help would be appreciated.

P.s :This is a crosspost from math.se :http://math.stackexchange.com/questions/187453/variants-of-isotopy-extensions, where I did not get any answer.