  This paper investigates the energy complexity of distributed graph problems
in multi-hop radio networks, where the energy cost of an algorithm is measured
by the maximum number of awake rounds of a vertex. Recent works revealed that
some problems, such as broadcast, breadth-first search, and maximal matching,
can be solved with energy-efficient algorithms that consume only $\text{poly}
\log n$ energy. However, there exist some problems, such as computing the
diameter of the graph, that require $\Omega(n)$ energy to solve. To improve
energy efficiency for these problems, we focus on a special graph class:
bounded-genus graphs. We present algorithms for computing the exact diameter,
the exact global minimum cut size, and a $(1 \pm\epsilon)$-approximate $s$-$t$
minimum cut size with $\tilde{O}(\sqrt{n})$ energy for bounded-genus graphs.
Our approach is based on a generic framework that divides the vertex set into
high-degree and low-degree parts and leverages the structural properties of
bounded-genus graphs to control the number of certain connected components in
the subgraph induced by the low-degree part.
