A Proof of a Conjecture on the Connected Domination Number

For a connected graph G, let γ(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma (G)$$\end{document} and γc(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{c}(G)$$\end{document} denote the domination number and the connected domination number, respectively. Let H be a graph obtained from a triangle abc by adding a pendant edge at a and a pendant path of length 3 at each of b and c. In 2014, Camby and Schaudt conjectured that for any connected {P9,C9,H}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{P_{9},C_{9},H\}$$\end{document}-free graph G, γc(G)≤2γ(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{c}(G)\le 2\gamma (G)$$\end{document}. In this paper, we settle the conjecture in the affirmative.