Disprove of a Conjecture on the Doubly Connected Domination Subdivision Number

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 γcc(G) is the minimum size of such a set. The doubly connected domination subdivision number sdγcc (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.