The ratio of the irredundance and domination number of a graph Original Research Article

Let γ(G) be the domination number and let ir(G) be the irredundance number of a simple graph G. The well-known inequality γ(G)⩽2ir(G) − 1 was obtained independently by Allan and Laskar (1978) and Bollobás and Cockayne (1979). For any tree T, Damaschke (1991) derived the sharper estimation 2γ(T) < 3 ir(T). Extending this result, we shall prove in this paper that 2γ(G) ⩽3 ir(G) for any block graph G and for any graph G with cyclomatic number μ(G)⩽2. In addition, we shall present examples which show that this estimation is not valid for μ(G)⩾3 in general.