## Araujo, Rafael and Costa, Eurinardo and Klein, Sulamita and Sampaio, Rudini and Souza, Ueverton S. - FPT algorithms to recognize well covered graphs

dmtcs:4905 - Discrete Mathematics & Theoretical Computer Science, April 2, 2019, vol. 21 no. 1, ICGT 2018
FPT algorithms to recognize well covered graphs

Authors: Araujo, Rafael and Costa, Eurinardo and Klein, Sulamita and Sampaio, Rudini and Souza, Ueverton S.

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). Moreover, 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 $\alpha(G)=n-vc(G)$ is the parameter. Contrasting with such coW[2]-hardness, we present an FPT algorithm to decide well coveredness when $\alpha(G)$ 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).

Source : oai:arXiv.org:1810.08276
Volume: vol. 21 no. 1, ICGT 2018
Published on: April 2, 2019
Submitted on: October 22, 2018
Keywords: Computer Science - Data Structures and Algorithms