On the lengths of symmetry breaking-preserving games on graphs

  • Authors:

  • F. Harary;W. Slany;O. Verbitsky

  • Affiliations:

  • Computer Science Department, New Mexico State University, Las Cruces, NM;Institut für Informationssysteme, Technische Universität Wien, Favoritenstr. 9, A-1040 Wien, Austria;Department of Mechanics & Mathematics, Lviv University, Universytetska 1, 79000 Lviv, Ukraine

  • Venue:
  • Theoretical Computer Science - Algorithmic combinatorial game theory
  • Year:
  • 2004

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Abstract

Given a graph G, we consider a game where two players, A and B, alternatingly, color edges of G in red and in blue, respectively. Let Lsym(G) be the maximum number of moves in which B is able to keep the red and the blue subgraphs isomorphic, if A plays optimally to destroy the isomorphism. This value is a lower bound for the duration of any avoidance game on G under the assumption that B plays optimally. We prove that if G is a path or a cycle of odd length n, then Ω(logn) ≤ Lsym(G) ≤ O(log2 n). The lower bound is based on relations with Ehrenfeucht-Fraïssé games from model theory. We also consider complete graphs and prove that Lsym(Kn) = O(1).