On the maximum orders of an induced forest, an induced tree, and a stable set

Let G be a connected graph, n the order of G, and f (resp. t) the maximum order of an induced forest (resp. tree) in G. We show that f − t is at most n − ⌠2√n − 1⌡ . In the special case where n is of the form a² + 1 for some even integer a ≥ 4, f − t is at most n − ⌠2√n − 1⌡ − 1. We also prove that these bounds are tight. In addition, letting α denote the stability number of G, we show that α − t is at most n + 1 − ⌠2√2n⌡; this bound is also tight.

Mots clés

Induced forest; induced tree; stability number; extremal graph theory