Dichotomizing [formula omitted]-vertex-critical [formula omitted]-free graphs for [formula omitted] of order four

Abstract

For every k≥1 and ℓ≥1, we prove that there is a finite number of k-vertex-critical (P2+ℓP1)-free graphs. This result establishes the existence of new polynomial-time certifying algorithms for deciding the k-colorability of (P2+ℓP1)-free graphs. Together with previous research, our result also implies the following characterization: There is a finite number of k-vertex-critical H-free graphs for H of order and for fixed k≥5 if and only if H is one of K4¯,P4,P2+2P1, or P3+P1. We also improve the recent known result that there is a finite number of k-vertex-critical (P3+P1)-free graphs for all k by showing that such graphs have at most 2k−1 vertices. We use this stronger result to exhaustively generate all k-vertex-critical (P3+P1)-free graphs for k≤7.