Coalition graphs of restrained dominating partitions of cycles

A set S of vertices of a graph G is a restrained dominating set if every vertex outside S has a neighbour in S and a neighbour outside S. An RD-partition of G is a vertex partition in which every part that is not a restrained dominating set forms, together with another such part, a restrained dominating set (an RD-coalition), and every part that is a restrained dominating set is a single vertex. The restrained coalition graph of an RD-partition has the parts as vertices, with two parts adjacent when they form an RD-coalition. Dobrynin, Glebov and Golmohammadi (Discrete Math. Lett. 17 (2026) 57–63) asked for a characterization of the coalition graphs generated by RD-partitions of cycles. We answer this question completely. For n = 3 the only such graph is the edgeless graph on three vertices. For every n ≥ 4 we determine exactly which graphs occur: they are K2, P3, K3, P4, C4, 2K2, P5, C5, P3 ∪ K2, 2P3 and 3K2, and for each of them we give the exact set of orders n for which it occurs. Thus exactly twelve graphs arise over all cycles. The proof is self-contained apart from the known bound RC(Cn) ≤ 6. It uses a local reformulation of restrained domination on cycles, which shows that every such coalition graph has maximum degree at most 2. We independently confirmed the result by exhaustive computer search.