Forbidden-subgraph classes of graphs with no long induced path are shown to have cop number at most about k/2 for (P_k,E)-free graphs and at most ceil(2p/3)+3 when the longest path has p vertices.
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Cops and Robbers on Graphs with Path Constraints
Forbidden-subgraph classes of graphs with no long induced path are shown to have cop number at most about k/2 for (P_k,E)-free graphs and at most ceil(2p/3)+3 when the longest path has p vertices.