Complexity of Stability

Authors Fabian Frei , Edith Hemaspaandra , Jörg Rothe

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Author Details

Fabian Frei
  • Department of Computer Science, ETH Zürich, Switzerland
Edith Hemaspaandra
  • Department of Computer Science, Rochester Institute of Technology, NY, USA
Jörg Rothe
  • Institut für Informatik, Heinrich-Heine-Universität Düsseldorf, Germany


We thank the anonymous referees for their careful reading of this paper and suggestions for improvement.

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Fabian Frei, Edith Hemaspaandra, and Jörg Rothe. Complexity of Stability. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Graph parameters such as the clique number, the chromatic number, and the independence number are central in many areas, ranging from computer networks to linguistics to computational neuroscience to social networks. In particular, the chromatic number of a graph (i.e., the smallest number of colors needed to color all vertices such that no two adjacent vertices are of the same color) can be applied in solving practical tasks as diverse as pattern matching, scheduling jobs to machines, allocating registers in compiler optimization, and even solving Sudoku puzzles. Typically, however, the underlying graphs are subject to (often minor) changes. To make these applications of graph parameters robust, it is important to know which graphs are stable for them in the sense that adding or deleting single edges or vertices does not change them. We initiate the study of stability of graphs for such parameters in terms of their computational complexity. We show that, for various central graph parameters, the problem of determining whether or not a given graph is stable is complete for Θ₂ᵖ, a well-known complexity class in the second level of the polynomial hierarchy, which is also known as "parallel access to NP."

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Complexity classes
  • Mathematics of computing → Extremal graph theory
  • Stability
  • Robustness
  • Complexity
  • Local Modifications
  • Colorability
  • Vertex Cover
  • Clique
  • Independent Set
  • Satisfiability
  • Unfrozenness
  • Criticality
  • DP
  • coDP
  • Parallel Access to NP


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