Orientation distance graphs (Q2725323)

The orientation distance between two orientations \(D\) and \(D'\) of a graph \(G\) is the minimum number of edges of \(G\) whose orientation needs to be reversed to transform \(D\) into an orientation isomorphic to \(D'\). The orientation distance graph with respect to \(G\) is a graph whose vertex set is a certain set of such orientations and in which two vertices are adjacent if and only if their orientation distance is one. If the vertex set is the set of all (pairwise non-isomorphic) orientations of \(G\), the orientation distance graph is denoted by \({\mathcal D}_0(G)\). The graph \({\mathcal D}_0(G)\) for \(G\) consisting of two non-isomorphic connected components is described. Further orientation distance graphs with respect to paths \(P_n\) with \(n\) vertices are studied. Every tree and every circuit is such a graph. If \(n\) is odd, then \({\mathcal D}_0(P_n)\) is bipartite. The graph \({\mathcal D}_0(P_n)\) for \(n\geq 4\) is Hamiltonian if and only if \(n\) is even.