Structural descriptions of (bull, house)-free and (bull, P5)-free graphs yield finiteness of k-critical instances for fixed k plus a short proof of perfect divisibility.
Discrete Appl
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Finiteness of k-vertex-critical graphs holds in (P4+ℓP1, chair)-free, (P4+ℓP1,P5,bull)-free, (P4+ℓP1,P5,cricket)-free, and more generally (P4+ℓP1,B4(m),B3(m)+)-free graphs, with χ ≤ ℓ+2 for (P4+ℓP1,K3)-free graphs.
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Structural description of (bull, house)-free graphs
Structural descriptions of (bull, house)-free and (bull, P5)-free graphs yield finiteness of k-critical instances for fixed k plus a short proof of perfect divisibility.
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Vertex-critical graphs in subfamilies of $(P_4+\ell P_1)$-free graphs
Finiteness of k-vertex-critical graphs holds in (P4+ℓP1, chair)-free, (P4+ℓP1,P5,bull)-free, (P4+ℓP1,P5,cricket)-free, and more generally (P4+ℓP1,B4(m),B3(m)+)-free graphs, with χ ≤ ℓ+2 for (P4+ℓP1,K3)-free graphs.