Abstract
Recognition of well-covered graphs is co-NP-complete, even when the graphs in question are k 1,4-free. On the other hand, very recently, an algorithm has been found which recognizes k 1,3-free well-covered graphs in polynomial time. Claw-free well-covered graphs having no 4-cycles have recently been characterized. In the present paper, we determine two different classes of claw-free well-covered graphs — those which are 4-connected and 4-regular as well as those which are 4-connected and planar.
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