Are there any special algorithms for maximum independent set of line graphs? Could this special case be in $\mathsf{P}$?

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Finding a maximum independent set in $L(G)$ is equivalent to finding a maximum matching in $G$. For more, and a fast polynomial-time algorithm that work for e.g., line graphs, see [1].

[1] Lozin, V.V. and Milanič, M., 2008. On finding augmenting graphs. Discrete Applied Mathematics, 156(13), pp.2517-2529.

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