For every k there is a graph whose cop number, independence number, and clique-cover number are all equal to k, and any graph with these equal for k≥3 contains induced cycles of every length from 3 to k+1.
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Cops and Robbers, Clique Covers, and Induced Cycles
For every k there is a graph whose cop number, independence number, and clique-cover number are all equal to k, and any graph with these equal for k≥3 contains induced cycles of every length from 3 to k+1.