Domination and irredundance in cubic graphs Original Research Article

The main purpose of this paper is to show that for every positive integer k there exists a connected cubic graph Hk whose upper irredundance number (IR(Hk)) and upper domination number (P(Hk)) satisfy IR(Hk) − T(Hk) ⩾ k, thus disproving a conjecture of Henning and Slater. It is known that for any n-vertex graph G with minimum degree δ, IR(G) ⩽ n − δ. We show that, in addition, if G is regular, then IR(G) ⩽ 12n. In both cases the extremal graphs are characterized and shown to satisfy F(G) = IR(G).