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On the computational complexity of defining sets

Hatami, H ; Sharif University of Technology | 2005

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  1. Type of Document: Article
  2. DOI: 10.1016/j.dam.2005.03.004
  3. Publisher: 2005
  4. Abstract:
  5. Suppose we have a family F of sets. For every S∈F, a set D⊆S is a defining set for (F,S) if S is the only element of F that contains D as a subset. This concept has been studied in numerous cases, such as vertex colorings, perfect matchings, dominating sets, block designs, geodetics, orientations, and Latin squares. In this paper, first, we propose the concept of a defining set of a logical formula, and we prove that the computational complexity of such a problem is Σ2-complete. We also show that the computational complexity of the following problem about the defining set of vertex colorings of graphs is Σ2-complete: INSTANCE: A graph G with a vertex coloring c and an integer k.QUESTION: If C(G) is the set of all χ(G)-colorings of G, then does (C(G),c) have a defining set of size at most k? Moreover, we study the computational complexity of some other variants of this problem. © 2005 Elsevier B.V. All rights reserved
  6. Keywords:
  7. Color ; Graph theory ; Mathematical operators ; Problem solving ; Set theory ; Complexity ; Defining sets ; Graph coloring ; Satisfiability ; Computational complexity
  8. Source: Discrete Applied Mathematics ; Volume 149, Issue 1-3 , 2005 , Pages 101-110 ; 0166218X (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/pii/S0166218X05000533