4K₁-free graph with the cop number $3$

The game of cops and robber is a two-player turn-based game played on a graph where the cops try to capture the robber. The cop number of a graph G, denoted by $c(G)$ is the minimum number of cops required to capture the robber. For a given class of graphs F, let c( F):=(F)|F∈ F\, and let Forb( F) denote the class of F-free graphs. We show that the complement of the Shrikhande graph is (4K₁,C_)-free for any ≥ 6 and has the cop number~$3$. This provides a counterexample for the conjecture proposed by Sivaraman (arxiv, 2019) which states that if G is C_-free for all ≥ 6, then c(G)≤ 2. This also gives a negative answer to the question posed by Turcotte (Discrete Math. 345:112660 (2022)) 112660. to check whether $c($Forb(pK₁))=p-2. Turcotte also posed the question to check whether $c($Forb(pK₁+K₂))≤ p+1, for p≥ 3. We prove that this result indeed holds. We also generalize this result for Forb(pK₁+qK₂). Motivated by the results of Baird et al. (Contrib. Discrete Math. 9:70--84 (2014)) and Turcotte and Yvon (Discrete Appl. Math. 301:74--98 (2021)), we define the upper threshold degree and lower threshold degree for a particular class of graphs and show some computational advantage to find the cop number using these.