Loose-versus-non-exhaustive partition conjecture for metric graphs

Fix a graph $\mathcal{G}$, a number $k\geq1$ and $p\in(0,\infty]$. Let $\mathcal{N}_p(\mathcal{P})$ denote the energy of a partition $\mathcal{P}$. A non-exhaustive rigid $k$-partition is a rigid partition whose parts do not exhaust $\mathcal{G}$, while an exhaustive loose $k$-partition is an exhaustive partition allowing the loose type of cuts considered in the paper. Define

$$\mathcal{N}_{k,p}^{\mathrm{loose}}(\mathcal{G})=\inf\left\{\mathcal{N}_p(\mathcal{P}):\mathcal{P}\text{ is an exhaustive loose $k$-partition of }\mathcal{G}\right\}.$$

Loose-versus-non-exhaustive partition conjecture. The infimum over non-exhaustive rigid $k$-partitions satisfies

$$\inf\left\{\mathcal{N}_p(\mathcal{P}):\mathcal{P}\text{ is a non-exhaustive but rigid $k$-partition of }\mathcal{G}\right\}\geq\mathcal{N}_{k,p}^{\mathrm{loose}}(\mathcal{G}).$$

The conjecture formalises the expectation that allowing loose exhaustive partitions is at least as effective as infimising over non-exhaustive rigid partitions. The paper motivates it through cutting and edge-lengthening surgery principles, but leaves the general statement for future work.