Regularity, matchings and Cameron–Walker graphs

Let G be a simple graph and let β(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta (G)$$\end{document} be the matching number of G. It is well-known that regI(G)⩽β(G)+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{reg}\,}}I(G) \leqslant \beta (G)+1$$\end{document}. In this paper we show that regI(G)=β(G)+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\,\mathrm{reg}\,}}I(G) = \beta (G)+1$$\end{document} if and only if every connected component of G is either a pentagon or a Cameron–Walker graph.