Coalescence and criticality of graphs

The domination number γ ( G ) of a graph G is the least number of vertices in a dominating set of G , and the lower irredundance number i r ( G ) is the least number of vertices in a maximal irredundant set of G . For each of these graph parameters, we establish bounds on the parameter of the coalescence of two graphs in terms of the parameters of the two respective graphs. These results are then utilised in the construction of graphs that are γ -critical but not i r -critical. Such graphs were not previously known to exist.