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)212]−⌊n−42⌋+⌊n−22⌋ nonisomorphic 3-minimal triangle-free n-vertex graphs when n≥7, where [x] denotes the nearest integer to x.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
