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A Dichotomy for Upper Domination in Monogenic Classes

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8881)


An upper dominating set in a graph is a minimal (with respect to set inclusion) dominating set of maximum cardinality. The problem of finding an upper dominating set is NP-hard for general graphs and in many restricted graph families. In the present paper, we study the computational complexity of this problem in monogenic classes of graphs (i.e. classes defined by a single forbidden induced subgraph) and show that the problem admits a dichotomy in this family. In particular, we prove that if the only forbidden induced subgraph is a \(P_4\) or a \(2K_2\) (or any induced subgraph of these graphs), then the problem can be solved in polynomial time. Otherwise, it is NP-hard.


  • Monogenic Classes
  • Restricted Graph Families
  • Maximum Cardinality
  • Private Neighbor
  • Polynomial-time Result

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Vadim Lozin: The author gratefully acknowledges support from DIMAP - the Center for Discrete Mathematics and its Applications at the University of Warwick, and from EPSRC, grant EP/L020408/1.

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AbouEisha, H., Hussain, S., Lozin, V., Monnot, J., Ries, B. (2014). A Dichotomy for Upper Domination in Monogenic Classes. In: Zhang, Z., Wu, L., Xu, W., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2014. Lecture Notes in Computer Science(), vol 8881. Springer, Cham.

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