Acyclic reducible bounds for outerplanar graphs

For a given graph G and a sequence ₁, ₂,..., ₙ of additive hereditary classes of graphs we define an acyclic (₁, ₂,...,Pₙ)-colouring of G as a partition (V₁, V₂,...,Vₙ) of the set V(G) of vertices which satisfies the following two conditions: 1. ${\displaystyle G\left[V_i\right]{\in }_i}$ 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 ${\displaystyle u\in V_i}$ and ${\displaystyle v\in V_j}$ is acyclic. A class R = ₁ ⊙ ₂ ⊙ ... ⊙ ₙ is defined as the set of the graphs having an acyclic (₁, ₂,...,Pₙ)-colouring. If ⊆ R, then we say that R is an acyclic reducible bound for . In this paper we present acyclic reducible bounds for the class of outerplanar graphs.