Stable and unstable graphs with total irredundance number zero

For a graph G = (V, E), a set S ⊆ V is total irredundant if for every vertex v ∈ V, the set N[v] - N[S - {v}] is not empty. The total irredundance number ir_t(G) is the minimum cardinality of a maximal total irredundant set of G. We study the structure of the class of graphs which do not have any total irredundant sets; these are called ir_t(0)-graphs. Particular attention is given to the subclass of ir_t(0)-graphs whose total irredundance number either does not change (stable) or always changes (unstable) under arbitrary single edge additions. Also studied are ir_t(0)-graphs which are either stable or unstable under arbitrary single edge deletions.