All Minimal Prime Extensions of Hereditary Classes of Graphs

The substitution composition of two disjoint graphs G1 and G2 is obtained by first removing a vertex x from G2 and then making every vertex in G1 adjacent to all neighbours of x in G2. Let F be a family of graphs defined by a set Z* of forbidden configurations. Giakoumakis [V. Giakoumakis, On the closure of graphs under substitution, Discrete Mathematics 177 (1997) 83–97] proved that F∗, the closure under substitution of F, can be characterized by a set Z∗ of forbidden configurations — the minimal prime extensions of Z. He also showed that Z∗ is not necessarily a finite set. Since substitution preserves many of the properties of the composed graphs, an important problem is the following: find necessary and sufficient conditions for the finiteness of Z∗. Giakoumakis [V. Giakoumakis, On the closure of graphs under substitution, Discrete Mathematics 177 (1997) 83–97] presented a sufficient condition for the finiteness of Z∗ and a simple method for enumerating all its elements. Since then, many other researchers have studied various classes of graphs for which the substitution closure can be characterized by a finite set of forbidden configurations. The main contribution of this paper is to completely solve the above problem by characterizing all classes of graphs having a finite number of minimal prime extensions. We then go on to point out a simple way for generating an infinite number of minimal prime extensions for all the other classes of F∗

Deciding whether there are infinitely many prime graphs with forbidden induced subgraphs

A homogeneous set of a graph G is a set X of vertices such that 2≤|X|V(G)| and no vertex in V(G)−X has both a neighbor and a non-neighbor in X. A graph is prime if it has no homogeneous set. We present an algorithm to decide whether a class of graphs given by a finite set of forbidden induced subgraphs contains infinitely many non-isomorphic prime graphs

On prime inductive classes of graphs

AbstractLet 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)