Graphs with unique Grundy dominating sets

Abstract Given a graph G consider a procedure of building a dominating set D in G by adding vertices to D one at a time in such a way that whenever vertex x is added to D there exists a vertex $y\in N_G[x]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:mrow mml:miy</mml:mi> mml:mo∈</mml:mo> mml:msub mml:miN</mml:mi> mml:miG</mml:mi> </mml:msub> mml:mrow mml:mo[</mml:mo> mml:mix</mml:mi> mml:mo]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> that becomes domina