Suppose that we have a set of disjoint, properly embedded closed curves $\left\{c_1,\cdots, c_k\right\}$ in a compact surface $\Sigma$ ,perhaps with nonempty boundary.
Then it is trivial that the existence of a proper subsurface $S\subseteq \Sigma$ each of whose boundary components is one of $c_i$ implies the linear dependence of the homology classes $\left\{[c_1],\cdots, [c_k]\right\}$. (The image of the boundary map $\partial[S]\in H_1(\Sigma)$ is a linear relation, for example.)
My question is, is the converse also true? That is, if the collection of curves $\left\{c_1,\cdots, c_k\right\}$ as above represents a linearly dependent elements in $H_1(\Sigma)$, can we find a proper subsurface $S$ whose boundary is a union of $c_i$'s?