On Minimum Dominating Sets in cubic and (claw,H)-free graphs

Given a graph $G=(V,E)$, $S\subseteq V$ is a dominating set if every $v\in V\setminus S$ is adjacent to an element of $S$. The Minimum Dominating Set problem asks for a dominating set with minimum cardinality. It is well known that its decision version is $NP$-complete even when $G$ is a claw-free graph. We give a complexity dichotomy for the Minimum Dominating Set problem for the class of $(claw, H)$-free graphs when $H$ has at most six vertices. In an intermediate step we show that the Minimum Dominating Set problem is $NP$-complete for cubic graphs

On the complexity of the minimum domination problem restricted by forbidden induced subgraphs of small size

We study the computational complexity of the minimum dominating set problem on graphs restricted by forbidden induced subgraphs. We give some dichotomies results for the problem on graphs defined by any combination of forbidden induced subgraphs with at most four vertices, implying either an NP-Hardness proof or a polynomial time algorithm. We also extend the results by showing that dominating set problem remains NP-hard even when the graph has maximum degree three, it is planar and has no induced claw, induced diamond, induced K4 nor induced cycle of length 4, 5, 7, 8, 9, 10 and 11.Fil: Lin, Min Chih. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Mizrahi, Michel Jonathan. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin