Almost claw‐free graphs

Abstract

Abstract We say that G is almost claw‐free if the vertices that are centers of induced claws ( K 1,3 ) 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 K 1,4 ‐free almost claw‐free graph is fully cycle extendable.