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 K4K4 nor induced cycle of length 4, 5, 7, 8, 9, 10 and 11.