Abstract
In this paper we study the algorithmic complexity of the following problems:
(1) | Given a vertex-colored graph X = (V,E,c), compute a minimum cardinality set of vertices S⊆ V such that no nontrivial automorphism of X fixes all vertices in S. A closely related problem is computing a minimum base S for a permutation group G ≤ Sym(n) given by generators, i.e., a minimum cardinality subset S ⊆ [n] such that no nontrivial permutation in G fixes all elements of S. Our focus is mainly on the parameterized complexity of these problems. We show that when k=|S| is treated as parameter, then both problems are | ||||
(2) | A notion related to fixing is individualization, which is a useful technique combined with the Weisfeiler-Leman procedure in algorithms for Graph Isomorphism. We explore the complexity of individualization: the problem of computing the minimum number of vertices we need to individualize in a given graph such that color refinement results in a graph with useful structural properties in the context of Graph Isomorphism and the Weisfeiler-Leman procedure. | ||||
- [1] . 1993. Fixed-parameter tractability and completeness IV: On completeness for
W[P] andPSPACE analogues. Annals of Pure and Applied LogicAnn. Pure Appl. Logic 73, 3 (1993), 235–276. Google ScholarCross Ref
- [2] . 2013. The parameterized complexity of fixpoint free elements and bases in permutation groups. In 8thInternational Symposium on Parameterized and Exact ComputationIPEC (Switzerland). Springer, 4–15. Google Scholar
Cross Ref
- [3] . 2016. The parameterized complexity of fixing number and vertex individualization in graphs. In 41stInternational Symposium on Mathematical Foundations of Computer ScienceMFCS. Leibniz-Zentrum für Informatik, 13:1–13:14. Google Scholar
Cross Ref
- [4] . 2015. On the power of color refinement. In 20thInternational Symposium Fundamentals of Computation TheoryFCT (Berlin). Springer, 339–350. Google Scholar
Cross Ref
- [5] . 2015. On Tinhofer’s linear programming approach to isomorphism testing. In 40thInternational Symposium on Mathematical Foundations of Computer ScienceMFCS (Berlin). Springer, 26–37. Google Scholar
Cross Ref
- [6] . 2015. Graph Isomorphism in quasipolynomial time. (2015). https://arxiv.org/abs/1512-03547.Google Scholar
- [7] . 1980. Random graph isomorphism. SIAM J. Comput. 9, 3 (1980), 628–635. Google Scholar
Digital Library
- [8] . 1983. Canonical labeling of graphs. In 15thAnnual ACM Symposium on Theory of ComputingSTOC. 171–183. Google Scholar
Digital Library
- [9] . 2011. Base size, metric dimension and other invariants of groups and graphs. Bulletin of the London Mathematical SocietyBull. London Math. Soc. 43, 2 (2011), 209–242. Google Scholar
Cross Ref
- [10] . 1992. Minimum bases for permutation groups: The greedy approximation. Journal of AlgorithmsJ. Algor. 13, 2 (1992), 297–306. Google Scholar
Digital Library
- [11] . 2006. Identifying graph automorphisms using determining sets. Electronic Journal of CombinatoricsElectr. J. Combin. 13 (2006), R78. http://www.combinatorics.org/ojs/index.php/eljc/article/view/v13i1r78.Google Scholar
Cross Ref
- [12] . 2006. Combinatorial Matrix Classes. Cambridge University Press.Google Scholar
Cross Ref
- [13] . 1992. An optimal lower bound on the number of variables for graph identification. CombinatoricaCombin. 12, 4 (1992), 389–410. Google Scholar
Cross Ref
- [14] . 2003. On the existence of subexponential parameterized algorithms. 67, 4 (2003), 789–807. Google Scholar
Digital Library
- [15] . 1996. Permutation Groups. Springer. Google Scholar
Cross Ref
- [16] . 2003. Cutting up is hard to do: The parameterised complexity of k-cut and related problems. Electronic Notes in Theoretical Computer Science 78 (2003), 209–222. Google Scholar
Cross Ref
- [17] . 2006. Destroying automorphisms by fixing nodes. Discrete MathematicsDiscrete Math. 306, 24 (2006), 3244–3252. Google Scholar
Digital Library
- [18] . 1998. A threshold of ln n for approximating set cover. J. ACM 45, 4 (1998), 634–652. Google Scholar
Digital Library
- [19] . 2004. Rigidity and separation indices of Paley graphs. Discrete MathematicsDiscrete Math. 289, 1-3 (2004), 157–161. Google Scholar
Digital Library
- [20] . 1999. Equivalence in finite-variable logics is complete for polynomial time. CombinatoricaCombin. 19, 4 (1999), 507–532. Google Scholar
Cross Ref
- [21] . 2006. Testing graph isomorphism in parallel by playing a game. In Automata, Languages and Programming, 33rd International Colloquium, ICALP 2006, Venice, Italy, July 10-14, 2006, Proceedings, Part I(
Lecture Notes in Computer Science , Vol. 4051), , , , and (Eds.). Springer, 3–14. Google ScholarDigital Library
- [22] . 2003. Completeness results for graph isomorphism. 66, 3 (2003), 549–566. Google Scholar
Digital Library
- [23] . 1979. A note on the graph isomorphism counting problem. 8, 3 (1979), 131–136. Google Scholar
Cross Ref
- [24] . 2014. Practical graph isomorphism, II. Journal of Symbolic ComputationJ. Symb. Comput. 60 (2014), 94–112. Google Scholar
Digital Library
- [25] . 2008. Distance paired domination numbers of graphs. Discrete MathematicsDiscrete Math. 308, 12 (2008), 2473–2483. Google Scholar
Digital Library
- [26] . 1994. Fractional isomorphism of graphs. Discrete MathematicsDiscrete Math. 132, 1-3 (1994), 247–265. Google Scholar
Digital Library
- [27] . 2005. Undirected st-connectivity in log-space. In 37thAnnual ACM Symposium on Theory of ComputingSTOC. 376–385. Google Scholar
Digital Library
- [28] . 2011. Isomorphism of (mis)labeled graphs. In 19thEuropean Symposium on AlgorithmsESA. Springer, Berlin, 370–381. Google Scholar
Cross Ref
- [29] . 1991. A note on compact graphs. Discrete Applied MathematicsDiscrete Appl. Math. 30, 2–3 (1991), 253–264. Google Scholar
Cross Ref
- [30] . 2001. Non-approximability results for optimization problems on bounded degree instances. In 33rdAnnual ACM Symposium on Theory of ComputingSTOC (New York). ACM, 453–461. Google Scholar
Digital Library
- [31] . 1982. Graph isomorphism problem. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta 118 (1982), 83–158.
Russian. Translation to English: [32]. Google Scholar - [32] . 1985. Graph isomorphism problem. Journal of Mathematical SciencesJ. Math. Sci. 29, 4 (1985), 1426–1481.
English translation of [31]. Google ScholarCross Ref
Index Terms
The Parameterized Complexity of Fixing Number and Vertex Individualization in Graphs
Recommendations
Parameterized complexity of vertex colouring
For a family F of graphs and a nonnegative integer k, F + ke and F - ke, respectively, denote the families of graphs that can be obtained from F graphs by adding and deleting at most k edges, and F + kv denotes the family of graphs that can be made into ...
Parameterized complexity of vertex deletion into perfect graph classes
Vertex deletion problems are at the heart of parameterized complexity. For a graph class F, the F-Deletion problem takes as input a graph G and an integer k. The question is whether it is possible to delete at most k vertices from G such that the ...
Uniquely colorable graphs up to automorphisms
Highlights- An extension of the concept of uniquely colorable graphs.
- Uniquely colorable ...
AbstractWe extend the concept of uniquely colorable graphs and say that a graph G is χ-iso-unique if for every two proper colorings c : V ( G ) → { 1 , … , χ ( G ) } and d : V ( G ) → { 1 , … , χ ( G ) } there exists an automorphism φ of G ...






Comments