Known Families of GraphsΒΆ
Here we introduce several known graphs that can be accessed in the PyRigi graph database.
Definition 39 (\(n\)-Frustum)
Assume that \(n\geq 3\). The graph \(G=(V,E)\) is called the \(n\)-Frustum if it is the Cartesian product \(G=C_n\,\square \, K_2\) of a cycle graph \(C_n\) on \(n\) vertices and the complete graph \(K_2\) on two vertices.
As a framework, the \(n\)-Frustum is typically realized as two regular \(n\)-sided polygons on circles centered in the origin of the Euclidean plane with radii \(r_1<r_2\). It has a nontrivial infinitesimal flex given by the rotation of the outer polygon while the inner polygon remains fixed. This infinitesimal flex does not extend to a continuous flex.
Definition 40 (Counterexample for the symmetry-adjusted Laman count with a free group action)
Let \(n\geq8\) be even, and let \(C_n\) denote the anti-clockwise rotation around the origin by \(2\pi/n\). Define a 4-regular graph \(H_n=(V_n,E_n)\) on the vertex set \(V_n=\{v_1,\dots,v_n\}\) such that each vertex \(v_i\) of \(H_n\) is exactly adjacent to \(v_{i+1}\), \(v_{i-1}\), \(v_{i+3}\) and \(v_{i-3}\) with \(v_{n+k}=v_k\) and \(v_{1-k}=v_{n+1-k}\) for \(k\in \{1,2,3\}\).
Define a framework \((H_n,p)\) on \(H_n\) realized on the unit circle in \(\mathbb{R}^2\) such that \(p(v_1)=C_np(v_n)\) and \(p(v_i)=C_np(v_{i-1})\) for all \(2\leq i\leq n\). For some vector \(t\) on the line from the origin to a vertex from \(\{v_1,\dots,v_n\}\), such a framework has a nontrivial infinitesimal motion \(m\) which satisfies the system of equations
where \(1\leq i\leq n\).
PyRigi: CnSymmetricFourRegular()
CnSymmetricFourRegular()
References: [LPS24]
Definition 41 (Counterexample for the symmetry-adjusted Laman count which contains a joint at the origin)
The previous example can be extended in the following way: add a vertex \(u\) and the vertices \(U_n=\{u_1,\dots,u_n\}\) to \(H_n\) from the previous definition. We also add the following edges to \(H_n\):
For all \(1\leq i\leq n\), add the edge \(\{u,u_i\}\).
For all \(1\leq i\leq n\), add the edge \(\{u_i,v_i\}\).
For all \(3\leq i\leq n\), add the edge \(\{u_i,u_{i-2}\}\). Also add the edges \(\{u_1,u_{n-1}\}\) and \(\{u_2,u_n\}\).
Denote the edge set created in this way by \(F_n\). This creates the graph \(G_n=(V_n\cup\{u\}\cup U_n,~E_n \cup F_n)\).
As a framework, this graph can be realized extending the realization \(p:V_n\rightarrow \mathbb{R}^2\) from before to \(G_n\): \(p(u)\) is placed at the origin, and the other \(n\) vertices \(\{p(u_1),\dots,p(u_n)\}\) are placed on a circle with radius \(r>0\) so that that \(p(u_1)=C_np(u_n)\) and \(p(u_i)=C_np(u_{i-1})\) for all \(2\leq i\leq n\). In this way, the new edges given in (iii) form two disjoint regular \(n/2\)-gons. The nontrivial infinitesimal motion from the previous example extends to a nontrivial infinitesimal motion of the new framework \((G_n,p)\) which rotates the two \(n/2\)-gons clockwise and anti-clockwise, respectively.
PyRigi: CnSymmetricFourRegularWithFixedVertex()
CnSymmetricFourRegularWithFixedVertex()
References: [LPS24]