On the stability of kernel bundles over chain-like curves

Let $C$ be a chain-like curve having $n$ smooth components and $n-1$ nodes, where $n\ge 2$. Let $E$ be a vector bundle on $C$ and $V\subseteq H^0(E)$ be a linear subspace generating $E$. We investigate the (semi)stability of the kernel bundle $M_{E,V}$ associated to $(E,V)$.