The Smallest Degree Sum that Yields Potentially \(C_{k}\)-graphical Sequences

We first prove that if \( G \) is a connected graph with \( n \) vertices and chromatic number \( \chi(G) = k \geq 2 \), then its independent domination number

\[i(G) \leq \left\lceil \frac{(k-1)}{k}n \right\rceil – (k-2).\]

This bound is tight and remains so for planar graphs. We then prove that the independent domination number of a diameter two planar graph on \( n \) vertices is at most \( \left\lceil \frac{n}{3} \right\rceil \).