Disprove of a Conjecture on the Doubly Connected Domination Subdivision Number
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A set S of vertices of a connected graph G is a doubly connected dominating set (DCDS) if every vertex not in S is adjacent to some vertex in S and the subgraphs induced by S and $$V-S$$
are connected. The doubly connected domination number $$\gamma _{cc}(G)$$
is the minimum size of such a set. The doubly connected domination subdivision number $$\hbox {sd}_{\gamma _{cc}}(G)$$
is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) to increase the doubly connected domination number. It was conjectured (Karami et al. in Mat Vesnic 64:232–239, 2012) that the doubly connected domination subdivision number of a connected planar graph is at most two. In this paper, we disprove this conjecture by showing that the doubly connected domination subdivision number of the regular icosahedron graph is three.
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