Abstract: | The detour order of a graph G, denoted by τ(G), is the order of a longest path in G. A subset S of V(G) is called a Pn-kernel of G if τ(G[S])≤n−1 and every vertex v∈V(G)−S is adjacent to an end-vertex of a path of order n−1 in G[S]. A partition of the vertex set of G into two sets, A and B, such that τ(G[A])≤a and τ(G[B])≤b is called an (a,b)-partition of G. In this paper we show that any graph with girth g has a Pn+1-kernel for every . Furthermore, if τ(G)=a+b, 1≤a≤b, and G has girth greater than , then G has an (a,b)-partition. |