Alikhani–Griswold, Section 4(3): connected graphs with domination root −4

We answer Section 4, item 3 of Alikhani and Griswold's On the Integer Domination Root Conjecture by constructing connected graphs with domination root −4 and unbounded order. Joining r copies of their 33-vertex graph through vertices adjacent to leaves gives a connected graph on 33r vertices whose domination polynomial is the rth power of the original polynomial. The root −4 has multiplicity exactly r, and the copies can be joined along a path with maximum degree seven. A mixed construction gives examples of every order at least 35, with maximum degree seven and cycle rank four. We give a complete set-bijection proof, an exact finite calculation for the base graph, and the scope of the corresponding Lean declarations.