Digraphs with isomorphic underlying and domination graphs: 4-cycles and pairs of paths

A domination graph of a digraph D, dom(D), is created using the vertex set of D, V(D). There is an edge uv in dom(D) whenever (u, z) or (v, z) is in the arc set of D, A(D), for every other vertex z is an element of V(D). For only some digraphs D has the structure of dom(D) been characterized. Examples of this are tournaments and regular digraphs. The authors have characterizations for the structure of digraphs D for which UG(D) = dom(D) or UG(D) congruent to dom(D). For example, when UG(D) congruent to dom(D), the only components of the complement of UG(D) are complete graphs, paths and cycles. Here, we determine values of i and j for which UG(D) congruent to dom(D) and UG(c)(D) = C-4 boolean OR P-i boolean OR P-j