On prime inductive classes of graphs

Abstract

Let H[G1,…,Gn] denote a graph formed from unlabelled graphs G1,…,Gn and a labelled graph H=({v1,…,vn},E) replacing every vertex vi of H by the graph Gi and joining the vertices of Gi with all the vertices of those of Gj whenever {vi,vj}∈E(H). For unlabelled graphs G1,…,Gn,H, let φH(G1,…,Gn) stand for the class of all graphs H[G1,…,Gn] taken over all possible orderings of V(H).A prime inductive class of graphs, I(B,C), is said to be a set of all graphs, which can be produced by recursive applying of φH(G1,…,G∣V(H)∣) where H is a graph from a fixed set C of prime graphs and G1,…,G∣V(H)∣ are either graphs from the set B of prime graphs or graphs obtained in the previous steps. Similar inductive definitions for cographs, k-trees, series–parallel graphs, Halin graphs, bipartite cubic graphs or forbidden structures of some graph classes were considered in the literature (Batagelj (1994) [1] Drgas-Burchardt et al. (2010) [6] and Hajós (1961) [10]).This paper initiates a study of prime inductive classes of graphs giving a result, which characterizes, in their language, the substitution closed induced hereditary graph classes. Moreover, for an arbitrary induced hereditary graph class P it presents a method for the construction of maximal induced hereditary graph classes contained in P and substitution closed.The main contribution of this paper is to give a minimal forbidden graph characterization of induced hereditary prime inductive classes of graphs. As a consequence, the minimal forbidden graph characterization for some special induced hereditary prime inductive graph classes is givenThere is also offered an algebraic view on the class of all prime inductive classes of graphs of the type I({K1},C).