A Characterization of Independent Domination Critical Graphs with a Cutvertex

Let \( i(G) \) denote the minimum cardinality of an independent dominating set for \( G \). A graph \( G \) is \( k \)-\( i \)-critical if \( i(G) = k \), but \( i(G + uv) < k \) for any pair of non-adjacent vertices \( u \) and \( v \) of \( G \). In this paper, we show that if \( G \) is a connected \( k \)-\( i \)-critical graph, for \( k \geq 3 \), with a cutvertex \( u \), then the number of components of \( G – u \), \( \omega(G – u) \), is at most \( k – 1 \) and there are at most two non-singleton components. Further, if \( \omega(G – u) = k – 1 \), then a characterization of such graphs is given.