Computational Complexity of Recognizing Well-Covered and Generalized Well-Covered Graphs

The core message of this article is to investigate the computational complexity of recognizing well-covered graphs and their generalizations, known as Wk graphs and Es graphs. The authors establish several complexity results, including showing that recognizing Wk graphs and shedding vertices are coNP-complete on well-covered graphs, determining the precise complexity of recognizing 1-extendable (Es) graphs as Θp2-complete, and providing a linear-time algorithm to decide if a chordal graph is 1-extendable.