6 problems

Let $G$ be a minimally 3-rigid graph, and let $G'$ be obtained from $G$ by a 1-extension of type E2Xs511 or E2Vs511. Let $\mathbf{c}_{3}(G)$ denote the three-dimensional realizatio…

Let $G$ be a minimally 3-rigid graph, and let $G'$ be obtained from $G$ by a 1-extension of type E1s63. Let $\mathbf{c}_{3}(G)$ denote the three-dimensional realization count. The…

Let $G$ be a minimally rigid graph, and let $G'$ be obtained from $G$ by a 1-extension of type E1a. Let $\mathbf{c}^{\circ}_{2}(G)$ denote the spherical realization count. The sphe…

Let $G$ be a minimally rigid graph, and let $G'$ be obtained from $G$ by a 1-extension of type E1a. Let $\mathbf{c}_{2}(G)$ denote the planar realization count. The planar E1a doub…

Edge-disjoint 1-factor conjecture. Some realization of $\mathrm{\pi}$ has $k$ edge-disjoint $1$-factors if and only if

Let $M_2(n)$ and $M_3(n)$ denote the maximum numbers of realizations in dimensions $2$ and $3$, respectively, among Laman graphs with $n$ vertices. For each $n\geqslant2$, let…