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In general, determining the domination numberand the feedback vertex numberof a claw-free graph (even a line graph) is NP-hard. In contrast, the situation becomes different for the complement of a line graph. In this paper, it is shown thatfor a claw-free  G  withand thus determining the domination number of the complement of a claw-free  G  withis polynomial, whereis the independence number of  G . Furthermore, if a graph  G  is not a star, has no isolated vertex and isolated edge, thenwhere  J ( G ) is the complement of the line graph  L ( G ) of  G . If a graph  G  is not a star, and has no isolated vertex,providedFor the case whenis also given. Thereby, we are able to show that determiningis polynomial.

Domination number and feedback vertex number of complements of line graphs