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An upper dominating set in a graph is a minimal (with respect to setinclusion) dominating set of maximum cardinality. The problem of finding anupper dominating set is generally NP-hard. We study the complexity of thisproblem in classes of graphs defined by finitely many forbidden inducedsubgraphs and conjecture that the problem admits a dichotomy in this family,i.e. it is either NP-hard or polynomial-time solvable for each class in thefamily. A helpful tool to study the complexity of an algorithmic problem onfinitely defined classes of graphs is the notion of boundary classes. However,none of such classes has been identified so far for the upper dominating setproblem. In the present paper, we discover the first boundary class for thisproblem and prove the dichotomy for classes defined by a single forbiddeninduced subgraph.

Upper Domination: towards a dichotomy through boundary properties