Dichotomizing k-vertex-critical H-free graphs for H of order four
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For k ≥ 3, we prove (i) there is a finite number of k-vertex-critical (P_2+ℓ P_1)-free graphs and (ii) k-vertex-critical (P_3+P_1)-free graphs have at most 2k-1 vertices. Together with previous research, these results imply the following characterization where H is a graph of order four: There is a finite number of k-vertex-critical H-free graphs for fixed k ≥ 5 if and only if H is one of K_4, P_4, P_2 + 2P_1, or P_3 + P_1. Our results imply the existence of new polynomial-time certifying algorithms for deciding the k-colorability of (P_2+ℓ P_1)-free graphs for fixed k.
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