On the planarity of iterated jump graphs (Q1841898)

In a base graph \(G\) a jump is the replacement of an edge of some edge-induced subgraph by another non-adjacent edge of \(G\). The \(r\)-jump graph of \(G\) has as vertices all \(r\)-edge edge-induced subgraphs of \(G\), and an edge for each jump. Taking the \(r\)-jump graph from the \(r\)-jump graph, and so on, from \(G\) yields the sequence of iterated \(r\)-jump graphs of \(G\), which may die out (become empty), may converge (become constant), or may diverge. It is shown that the sequence of iterated \(1\)-jump graphs of \(G\) converges iff their genus converges iff they are all planar iff \(G\) is either the cycle \(C_5\) or the corona cor\((K_3)\) (\(K_3\) with one pending edge added to each vertex). For \(r=2\) the same series of equivalent properties hold iff \(G\) is the cycle \(C_4\).