Path Partitions and Pn-free Sets

The detour order τ(G) of a graph G is the order of a longest path of G. A partition (A, B) of V is called an (a, b)-partition of G if τ(G[A]) ≤ a and τ(G[B]) ≤ b. The Path Partition Conjecture is the following: y graph G, with detour order τ(G) = a + b, there exists an (a, b)-partition of G. roduce and examine a conjecture which is possibly stronger: If M is a maximum Pn+1-free set of vertices of G, with n < τ(G), then τ(G − M) ≤ τ(G)− n.