Nonplanar Sequences of Iterated \(r\)-Jump Graphs

Abstract

For a graph \(G\) of size \(m \geq 1\) and edge-induced subgraphs \(F\) and \(H\) of size \(r\) (\(1 \leq r \leq m\)), the subgraph \(Z\) is said to be obtained from \(F\) by an edge jump if there exist four distinct vertices \(u, v, w\), and \(x\) in \(G\) such that \(uv \in E(F)\), \(wx \in E(G) – E(F)\), and \(H = F – uv + wx\). The minimum number of edge jumps required to transform \(F\) into \(H\) is the jump distance from \(F\) to \(H\). For a graph \(G\) of size \(m \geq 1\) and an integer \(r\) with \(1 \leq r \leq m\), the \(r\)-jump graph \(J_r(G)\) is that graph whose vertices correspond to the edge-induced subgraphs of size \(r\) of \(G\) and where two vertices of \(J_r(G)\) are adjacent if and only if the jump distance between the corresponding subgraphs is \(1\). For \(k \geq 2\), the \(k\)th iterated jump graph \(J^k(G)\) is defined as \(J_r(J^{k-1}_{r}(G))\), where \(J^1_r(G) = J_r(G)\). An infinite sequence \(\{G_i\}\) of graphs is planar if every graph \(G_i\) is planar; while the sequence \(\{G_i\}\) is nonplanar otherwise. It is shown that if \(\{J^k_2(G)\}\) is a nonplanar sequence, then \(J^k_2(G)\) is nonplanar for all \(k \geq 3\) and there is only one graph \(G\) such that \(J^2_2(G)\) is planar. Moreover, for each integer \(r \geq 3\), if \(G\) is a connected graph of size at least \(r + 2\) for which \(\{J^k_r(G)\}\) is a nonplanar sequence, then \(J^k_r(G)\) is nonplanar for all \(k \geq 3\).