Abstract

Given a directed graph G=(V(G),A(G)), a subset X of V(G) is an interval of G provided that for any a,bX and xV(G)-X, (a,x)A(G) if and only if (b,x)A(G), and similarly for (x,a) and (x,b). For example, , {x} (xV(G)) and V(G) are intervals of G, called trivial intervals. A directed graph is indecomposable if all its intervals are trivial; otherwise, it is decomposable. An indecomposable directed graph G is then critical if for each xV(G), G(V(G)-{x}) is decomposable and if there are x≠yV(G) such that G(V(G)-{x,y}) is indecomposable. A generalization of the lexicographic sum is introduced to describe a process of construction of the critical and infinite directed graphs. It follows that for every critical and infinite directed graph G, there are x≠yV(G) such that G and G(V(G)-{x,y}) are isomorphic. It is then deduced that if G is an indecomposable and infinite directed graph and if there is a finite subset F of V(G) such that |F|2 and G(V(G)-F) is indecomposable, then there are x≠yV(G) such that G(V(G)-{x,y}) is indecomposable.

Keywords: Indecomposable; Critical; Generalized lexicographic sum and quotient

Mathematical subject codes: 05C20; 05C75

Article Outline

1. Introduction

1.1. Indecomposable graphs

1.2. Generalized quotient and generalized lexicographic sum

2. Indecomposability graph

3. Characterization of the critical and infinite graphs

4. Indecomposable and infinite graphs

5. Particular critical and infinite graphs

5.1. The elements of

5.2. The elements of

References

Supported by the France-Tunisia cooperation CNRS/DGRST.