Structural properties and characterizations of $\mathbf{W}_{p}$ class

Abstract

We establish new characterizations of graphs belonging to the $\mathbf{W}_p$ class. In addition, we characterize locally triangle-free $\alpha$-critical graphs in this class. As a consequence, our results yield a partial answer to a question raised by Plummer \cite{Plum93} in the case $p=2$.