On the planarity of iterated jump graphs

Abstract

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 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⩾2 and r⩾1, the kth iterated r-jump graph J r k(G) is defined as J r(J r k−1(G)) , where J r 1(G)=J r(G) . An infinite sequence {G i} of graphs is planar if every graph G i is planar. All graphs G for which {J r k(G)} ( r=1,2) is planar are determined and it is shown that if the sequence {J r k(G)} is nonplanar, then lim k→∞ gen( J r k(G))=∞ , where gen(G) denotes the genus of a graph G.