A class of planar well‐covered graphs with girth four
Michael R. Pinter · Journal of Graph Theory · 1995
Abstract A well‐covered graph is a graph in which every maximal independent set is a maximum independent set; Plummer introduced the concept in a 1970 paper. The notion of a 1‐well‐covered graph was introduced by Staples in her 1975 dissertation: a well‐covered graph G is 1‐well‐covered if and only if G ‐ v is also well covered for every point v in G. Except for K2 and C5, every 1‐well‐covered graph contains triangles or 4‐cycles. We show that all planar 1‐well‐covered graphs of girth 4 belong to a specific infinite family, and we give a characterization of this family. © 1995 John Wiley & Sons, Inc.