Almost claw-free graphs

@article{Ryjcek1994AlmostCG,
  title={Almost claw-free graphs},
  author={Zdenek Ryj{\'a}cek},
  journal={Journal of Graph Theory},
  year={1994},
  volume={18},
  pages={469-477}
}
We say that G is almost claw-free if the vertices that are centers of induced claws (K,J in G are independent and their neighborhoods are 2-dominated. Clearly, every claw-free graph is almost claw-free. It is shown that (i) every even connected almost claw-free graph has a perfect matching and (ii) every nontrivial locally connected Kl,4-free almost claw-free graph is fully cycle extendable.