Critical [formula omitted]-free graphs

Abstract

Given two graphs H1 and H2, a graph is (H1,H2)-free if it contains no induced subgraph isomorphic to H1 or H2. Let Pt and Ct be the path and the cycle on t vertices, respectively. A banner is the graph obtained from a C4 by adding a new vertex and making it adjacent to exactly one vertex of the C4. In this paper, we show that there are finitely many k-critical (P6,banner)-free graphs for k=4 and k=5. For k=4, we characterize all such graphs. Our results generalize previous results on k-critical (P6,C4)-free graphs for k=4 and k=5.