Independence Number of Maximal Planar Graphs

It is known that for a maximal planar graph \( G \) with order \( n \geq 4 \), the independence number satisfies \( \frac{n}{4} \leq \alpha(G) \leq \frac{2n-4}{3} \). We show that the lower bound is sharp and characterize the extremal graphs for \( n \leq 12 \). For the upper bound, we characterize the extremal graphs of all orders.

The independence number \( \alpha(G) \) of a graph \( G \) is the size of the largest independent set. This parameter is difficult to determine in general, but can be bounded on various graph classes. This paper considers planar and maximal planar graphs.