Galois quotients of metric graphs and invariant linear systems

For a map $\varphi : \varGamma \rightarrow \varGamma^{\prime}$ between metric graphs and an isometric action on $\varGamma$ by finite group $K$, $\varphi$ is a $K$-Galois covering on $\varGamma^{\prime}$ if $\varphi$ is a morphism, the degree of $\varphi$ coincides with the order of $K$ and $K$ induces a transitive action on every fibre. We prove that for a metric graph $\varGamma$ with an isometric action by finite group $K$, there exists a rational map, from $\varGamma$ to a tropical projective space, which induces a $K$-Galois covering on the image. By using this fact, we also prove that for a hyperelliptic metric graph without one valent points and with genus at least two, the invariant linear system of the hyperelliptic involution $\iota$ of the canonical linear system, the complete linear system associated to the canonical divisor, induces an $\langle \iota \rangle$-Galois covering on a tree. This is an analogy of the fact that a compact Riemann surface is hyperelliptic if and only if the canonical map, the rational map induced by the canonical linear system, is a double covering on a projective line $\boldsymbol{P}^1$.