On the planarity of jump graphs

Abstract

For a graph G of size m⩾1 and edge-induced subgraphs F and H of size k ( 1⩽k⩽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 k-jump distance from F to H. For a graph G of size m⩾1 and an integer k with 1⩽k⩽m, the k-jump graph J k(G) is that graph whose vertices correspond to the edge-induced subgraphs of size k of G and where two vertices of J k(G) are adjacent if and only if the k-jump distance between the corresponding subgraphs is 1. All connected graphs G for which J 2(G) is planar are determined.