Abstract
In a graph G, a module is a vertex subset M such that every vertex outside M is adjacent to all or none of M. A graph G is prime if phi, the single-vertex sets, and V (G) are the only modules in G. A prime graph G is k-minimal if there is some k-set U of vertices such that no proper induced subgraph of G containing U is prime.
Cournier and Ille in 1998 characterized the 1-minimal and 2-minimal graphs. We characterize 3-minimal triangle-free graphs. As a corollary, we show that there are exactly [(n-1)(2)/12] - [n-4/2] + [n-2/2] nonisomorphic 3-minimal triangle-free n-vertex graphs when n >= 7, where [x] denotes the nearest integer to x. (C) 2014 Elsevier B.V. All rights reserved.