3-minimal triangle-free graphs

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 ϕ , 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 .