Critical graphs upon multiple edge subdivision

A subset D of V(G) is a dominating set of a graph G if every vertex of V(G) - D has at least one neighbour in D ; let the domination number gamma (G) be the minimum cardinality among all dominating sets in G. We say that a graph G is gamma-q-critical if subdividing any q edges results in a graph with domination number greater than gamma(G) and there exists a set of q - 1 edges such that subdividing these edges results in a graph with domination number gamma(G). In this paper we consider mainly gamma-q-critical trees and give some general properties of gamma-q-critical graphs; in particular, we characterize those trees T that are gamma-(n (T)-1)-critical. We also characterize gamma-2-critical trees T with sd(T) = 2 and gamma-3-critical trees T with sd(T) = 3 , where the domination subdivision number sd(G) of a graph G is the minimum number of edges which must be subdivided (where each edge can be subdivided at most once) to construct a graph with domination number greater than gamma(G).