Dichotomizing k-vertex-critical H-free graphs for H of order four

For every k ≥ 1 and ℓ , we prove that there is a finite number of -vertex-critical ( P 2 + ) -free graphs. This result establishes the existence new polynomial-time certifying algorithms for deciding -colorability Together with previous research, our also implies following characterization: There H graphs order fixed 5 if only one K 4 ¯ or 3 . We improve recent known all by showing such have at most − vertices. use this stronger to exhaustively generate ≤ 7