Coalescence and criticality of graphs

The domination number γ(G)γ(G) of a graph GG is the least number of vertices in a dominating set of GG, and the lower irredundance number ir(G)ir(G) is the least number of vertices in a maximal irredundant set of GG. 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 γγ-critical but not irir-critical. Such graphs were not previously known to exist.