Independent arcs of acyclic orientations of complete r-partite graphs

Suppose D is an acyclic orientation of a graph G. An arc of D is said to be independent if its reversal results another acyclic orientation. Denote i(D) the number of independent arcs in D and N(G) = {i(D) : D is an acyclic orientation of G}. Also, let imin(G) be the minimum of N(G) and imax(G) the maximum. While it is known that imin(G) = |V (G)| − 1 for any connected graph G, the present paper determines imax(G) for complete r-partite graphs G. We also determine N(G) for balanced complete r-partite graphs G, and find some complete r-partite graphs G whose N(G) is a set of consecutive integers. As a consequence, we answer a question of West that there exist graphs G whose N(G) are not a set of consecutive integers.