Some Graphs with Double Domination Subdivision Number Three

A subset \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${S \subseteq V(G)}$$\end{document} is a double dominating set of G if S dominates every vertex of G at least twice. The double domination numberdd(G) is the minimum cardinality of a double dominating set of G. The double domination subdivision number sddd(G) is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the double domination number. Atapour et al. (Discret Appl Math, 155:1700–1707, 2007) posed an open problem: Prove or disprove: let G be a connected graph with no isolated vertices, then 1 ≤ sddd(G) ≤ 2. In this paper, we disprove the problem by constructing some connected graphs with no isolated vertices and double domination subdivision number three.