Abstract
A subset $${S \subseteq V(G)}$$ is a double dominating set of G if S dominates every vertex of G at least twice. The double domination number dd(G) is the minimum cardinality of a double dominating set of G. The double domination subdivision number sd dd (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 ≤ sd dd (G) ≤ 2. In this paper, we disprove the problem by constructing some connected graphs with no isolated vertices and double domination subdivision number three.
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