On the planarity of iterated jump graphs Original Research Article

For a graph G of size m⩾1 and edge-induced subgraphs F and H of size r (1⩽r⩽m), the subgraph H 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∈E(F), wx∈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⩾1 and an integer r with 1⩽r⩽m, the r-jump graph Jr(G) is that graph whose vertices correspond to the edge-induced subgraphs of size r of G and where two vertices of Jr(G) are adjacent if and only if the jump distance between the corresponding subgraphs is 1. For k⩾2 and r⩾1, the kth iterated r-jump graph Jrk(G) is defined as Jr(Jrk−1(G)), where Jr1(G)=Jr(G). An infinite sequence {Gi} of graphs is planar if every graph Gi is planar. All graphs G for which {Jrk(G)} (r=1,2) is planar are determined and it is shown that if the sequence {Jrk(G)} is nonplanar, then limk→∞ gen(Jrk(G))=∞, where gen(G) denotes the genus of a graph G.