FPT algorithms to recognize well covered graphs
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Given a graph G, let vc(G) and vc^+(G) be the sizes of a minimum and a maximum minimal vertex covers of G, respectively. We say that G is well covered if vc(G)=vc^+(G) (that is, all minimal vertex covers have the same size). Determining if a graph is well covered is a coNP-complete problem. In this paper, we obtain O^*(2^vc)-time and O^*(1.4656^vc^+)-time algorithms to decide well coveredness, improving results of Boria et. al. (2015). MoreoverBesides that, using crown decomposition, we show that such problems admit kernels having linear number of vertices. In 2018, Alves et. al. (2018) proved that recognizing well covered graphs is coW[2]-hard when the independence number α(G)=n-vc(G) is the parameter. Contrasting with such coW[2]-hardness, we present an FPT algorithm to decide well coveredness when α(G) and the degeneracy of the input graph G are aggregate parameters. Finally, we use the primeval decomposition technique to obtain a linear time algorithm for extended P_4-laden graphs and (q,q-4)-graphs, which is FPT parameterized by q, improving results of Klein et al (2013).
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