The structure of well-covered graphs with no cycles of length 4 Original Research Article

Let image be the set of well-covered graphs with no cycles of length 4. The main result is that if image then image can be partitioned, using an equivalence relation, into subsets image such that: (i) each image is well-covered; (ii) image; and (iii) the vector space of the well-covered weightings of G is the direct sum of the vector spaces of the well-covered weightings of the image, each of which has dimension 1. Our second result is that the problem of determining whether an edge of a graph is incident with two vertices in the same equivalence class is NP-complete. We give a forbidden co-stable subgraph characterization of graphs in image. Finally, we prove that graphs in image of bounded maximum generalized degree can be recognized in polynomial time.