Stable and Unstable Graphs With Total Irredundance Number Zero

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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 irt(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 irt(0)-graphs. Particular attention is given to the subclass of irt(0)-graphs whose total irredundance number either does not change (stable) or always changes (unstable) under arbitrary single edge additions. Also studied are irt(0)-graphs which are either stable or unstable under arbitrary single edge deletions.

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