On prime inductive classes of graphs

• Authors: Ewa Drgas-Burchardt
• Venue: European Journal of Combinatorics
• Year: 2011

Let H[G"1,...,G"n] denote a graph formed from unlabelled graphs G"1,...,G"n and a labelled graph H=({v"1,...,v"n},E) replacing every vertex v"i of H by the graph G"i and joining the vertices of G"i with all the vertices of those of G"j whenever {v"i,v"j}@?E(H). For unlabelled graphs G"1,...,G"n,H, let @f"H(G"1,...,G"n) stand for the class of all graphs H[G"1,...,G"n] 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 @f"H(G"1,...,G"|"V"("H")"|) where H is a graph from a fixed set C of prime graphs and G"1,...,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 Hajos (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 given There is also offered an algebraic view on the class of all prime inductive classes of graphs of the type I({K"1},C).