Domination and Independent Domination Numbers of Graphs

Let \(X\) be a graph and let \(\alpha(X)\) and \(\alpha'(X)\) denote the domination number and independent domination number of \(X\), respectively. We show that for every triple \((m,k,n)\), \(m \geq 5\), \(2 \leq k \leq m\), \(n > 1\), there exist \(m\)-regular \(k\)-connected graphs \(X\) with \(\alpha'(X) – \alpha(X) > n\). The same also holds for \(m = 4\) and \(k \in \{2,4\}\).