4 problems

Finiteness conjecture. For each odd $k\in \mathbb{Z}^+$, there exist only finitely many connected graphs with metamour period $k$.

Partial-short reduction conjecture. Starting with a weak graph, identify vertices having the least number of degree-$2$ neighbours, then among those select vertices of least degree…

Let $G$ be an arithmetical structure with associated quantities $\boldsymbol{d},\boldsymbol{r}$ and let $G'$ be obtained by the generalized star-clique operation, with associated q…

Let $X$-replacement and $V$-replacement be graph operations used to construct graphs in the three-dimensional rigidity setting, with additional conditions imposed on $V$-replacemen…