The ratio of the irredundance number and the domination number for block-cactus graphs

Let γ(G) and ir(G) denote the domination number and the irredundance number of a graph G, respectively. Allan and Laskar [1] and Bollobás and Cockayne [2] proved independently that γ(G) < 2 ir(G) for any graph G. For a tree T , Damaschke [4] obtained the sharper estimation 2γ(T ) < 3 ir(T ). Extending Damaschke’s result, Volkmann [11] proved that 2γ(G) ≤ 3 ir(G) for any block graph G and for any graph G with cyclomatic number μ(G) ≤ 2. Volkmann [11] also conjectured that 5γ(G) < 8 ir(G) for any cactus graph. In this article we show that if G is a blockcactus graph having π(G) induced cycles of length 2 (mod4), then γ(G)(5π(G)+4) ≤ ir(G)(8π(G) + 6). This result implies the inequality 5γ(G) < 8 ir(G) for a blockcactus graph G, thus proving the above conjecture. J. Graph Theory 29 (1998), 139-149