Domination in quadrangle-free Helly graphs (Q1842416)

The \(r\)-domination problem is considered on Helly graphs without quadrangles, i.e., without induced cycles of length 4. Alongside of other results, we demonstrate the polynomial-time solvability of the \(r\)- domination problem on bridged graphs without isometric \(k\)-suns \(S_ k\) \((k\geq 3)\) and bridged graphs without induced subgraphs \(S_ 3\), \(\overline S_ 3\). Recall that even the dominating set problem is NP- complete on all quadrangle-free Helly graphs, see the author, \textit{K. F. Prisakar} and \textit{V. D. Chepoj} [Diskretn. Mat. 4, No. 4, 67-73 (1992; Zbl 0804.05067)].