3 problems

Let $\mathcal{M}=(G,In,Out,Leak)$ be a linear compartment model with input, output, and leak in compartment $1$ only, so that $In=Out=Leak=\{1\}$, and suppose that $G$ is a bidirec…

Let $G$ be a graph on $n-1$ vertices, and let $k\to l$ be an edge in $G$. Construct a graph $G'$ on $n$ vertices by subdividing the edge $k\to l$: add vertex $n$ and replace the ed…

Let $G$ be a graph with $n$ vertices, $2n-2$ edges, and an exchange with $i$. The collapsed graph $G'$ is obtained by identifying vertices $1$ and $i$. Assume that $G'$ has $2n-4$…