All minimal prime extensions of hereditary classes of graphs

Abstract

The substitution composition of two disjoint graphs G 1 and G 2 is obtained by first removing a vertex x from G 2 and then making every vertex in G 1 adjacent to all neighbours of x in G 2. 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 configurationsthe 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 * .

Substitution-closed pattern classes

“…Similar notions have recently been studied in graph theory [3,[8][9][10][12][13][14][15]. For graphs there are natural analogues of the notions of 'basis', 'interval' and 'simple' and the substitution closure has been a tool in graph theory since Lovász's work [11] on perfect graphs.…”

“…For graphs there are natural analogues of the notions of 'basis', 'interval' and 'simple' and the substitution closure has been a tool in graph theory since Lovász's work [11] on perfect graphs. In the last decade much work has been done on determining the graphs of finite type and this recently culminated in a complete characterisation [10]. There is a connection between permutations and graphs in that every permutation determines a graph with the set of permuted points as vertices and the inversions as edges; then, in the terminology of [8][9][10], intervals correspond to modules and simple permutations to prime graphs.…”

“…In the last decade much work has been done on determining the graphs of finite type and this recently culminated in a complete characterisation [10]. There is a connection between permutations and graphs in that every permutation determines a graph with the set of permuted points as vertices and the inversions as edges; then, in the terminology of [8][9][10], intervals correspond to modules and simple permutations to prime graphs. However two different permutations (such as 4123 and 2341) can give rise to the same graph (a star graph on 4 vertices in this case, sometimes called a 3-claw) and so the connection is quite weak.…”

Prime bound of a graph

“…We complete the section with another equivalence relation induced by the modular decomposition tree. It is used by Giakoumakis and Olariu [10] to construct a minimal prime extension of a graph. Given a graph G, consider the equivalence relation…”

“…A prime extension H of G is minimal [14,9,18,1,10] if for every W ⊊ V (H) such that H[W ] is prime, H[W ] does not admit an induced subgraph isomorphic to G. Given a graph G, a prime p(G)-extension of G is clearly minimal. By Theorem 1, p(G) ≤ ⌈log 2 ( V (G) + 1)⌉.…”