Acyclic reducible bounds for outerplanar graphs

For a given graph G and a sequence P1,P2, . . . ,Pn of additive hereditary classes of graphs we define an acyclic (P1,P2, . . . ,Pn)colouring of G as a partition (V1, V2, . . . , Vn) of the set V (G) of vertices which satisfies the following two conditions: 1. G[Vi] ∈ Pi for i = 1, . . . , n, 2. for every pair i, j of distinct colours the subgraph induced in G by the set of edges uv such that u ∈ Vi and v ∈ Vj is acyclic. A class R = P1 P2 · · · Pn is defined as the set of the graphs having an acyclic (P1,P2, . . . ,Pn)-colouring. If P ⊆ R, then we say that R is an acyclic reducible bound for P. In this paper we present acyclic reducible bounds for the class of outerplanar graphs.