Graph Amalgamation Under Logical Constraints

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11159)


We say that a graph G is an H-amalgamation of graphs \(G_1\) and \(G_2\) if G can be obtained by gluing \(G_1\) and \(G_2\) along isomorphic copies of H. In the amalgamation recognition problem we are given connected graphs \(H,G_1,G_2,G\) and the goal is to determine whether G is an H-amalgamation of \(G_1\) and \(G_2\). Our main result states that amalgamation recognition can be solved in time \(2^{O(\varDelta \cdot t)}\cdot n^{O(t)}\) where \(n,t,\varDelta \) are the number of vertices, the treewidth and the maximum degree of G respectively.

We generalize the techniques used in our algorithm for H-amalgamation recognition to the setting in which some of the graphs \(H,G_1,G_2,G\) are not given explicit at the input but are instead required to satisfy some topological property expressible in the counting monadic second order logic of graphs (CMSO logic). In this way, when restricting ourselves to graphs of constant treewidth and degree our approach generalizes certain algorithmic decomposition theorems from structural graph theory from the context of clique-sums to the context in which the interface graph H is given at the input.


Graph amalgamation CMSO logic Logical constraints 



The author thanks Michael Fellows for many valuable comments. This work was supported by the Bergen Research Foundation.


  1. 1.
    Adler, I., Grohe, M., Kreutzer, S.: Computing excluded minors. In: Proceedings of SODA 2008, pp. 641–650. SIAM (2008)Google Scholar
  2. 2.
    Bojańczyk, M., Pilipczuk, M.: Definability equals recognizability for graphs of bounded treewidth. In Procedings of LICS 2016, pp. 407–416. ACM (2016)Google Scholar
  3. 3.
    Comon, H., et al.: Tree Automata Techniques and Applications (2007)Google Scholar
  4. 4.
    Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Courcelle, B.: The monadic second-order logic of graphs xii: planar graphs and planar maps. Theor. Comput. Sci. 237(1), 1–32 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Courcelle, B.: The monadic second-order logic of graphs xiii: graph drawings with edge crossings. Theor. Comput. Sci. 244(1–2), 63–94 (2000)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Courcelle, B., Engelfriet, J.: Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach, vol. 138. Cambridge University Press, Cambridge (2012)CrossRefGoogle Scholar
  8. 8.
    Courcelle, B., Oum, S.: Vertex-minors, monadic second-order logic, and a conjecture by Seese. J. Combina. Theory, Ser. B 97(1), 91–126 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    de Oliveira Oliveira, M.: On supergraphs satisfying CMSO properties. In: Proceedings of the 26th Annual Conference on Computer Science Logic (CSL 2017). LIPIcs, vol. 82, pp. 33:1–33:15 (2017)Google Scholar
  10. 10.
    Demaine, E.D., Fomin, F.V., Hajiaghayi, M., Thilikos, D.M.: Subexponential parameterized algorithms on bounded-genus graphs and h-minor-free graphs. J. ACM (JACM) 52(6), 866–893 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Diestel, R.: A separation property of planar triangulations. J. Graph Theory 11(1), 43–52 (1987)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Elberfeld, M.: Context-free graph properties via definable decompositions. In: Proceedings of the 25th Conference on Computer Science Logic (CSL 2016). LIPIcs, vol. 62, pp. 17:1–17:16 (2016)Google Scholar
  13. 13.
    Flum, J., Frick, M., Grohe, M.: Query evaluation via tree-decompositions. J. ACM (JACM) 49(6), 716–752 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gross, J.L.: Genus distribution of graph amalgamations: self-pasting at root-vertices. Australas. J. Combin. 49, 19–38 (2011)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hilton, A.J., Johnson, M., Rodger, C.A., Wantland, E.B.: Amalgamations of connected k-factorizations. J. Combin. Theory, Ser. B 88(2), 267–279 (2003)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Johnson, C.R., McKee, T.A.: Structural conditions for cycle completable graphs. Discret. Math. 159(1–3), 155–160 (1996)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kriz, I., Thomas, R.: Clique-sums, tree-decompositions and compactness. Discret. Math. 81(2), 177–185 (1990)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Leach, C.D., Rodger, C.: Hamilton decompositions of complete multipartite graphs with any 2-factor leave. J. Graph Theory 44(3), 208–214 (2003)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lovász, L.: Graph minor theory. Bull. Am. Math. Soc. 43(1), 75–86 (2006)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Makowsky, J.A.: Coloured tutte polynomials and Kauffman brackets for graphs of bounded tree width. Discret. Appl. Math. 145(2), 276–290 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Makowsky, J.A., Marino, J.P.: Farrell polynomials on graphs of bounded tree width. Adv. Appl. Math. 30(1–2), 160–176 (2003)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Makowsky, J.A., Rotics, U., Averbouch, I., Godlin, B.: Computing graph polynomials on graphs of bounded clique-width. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 191–204. Springer, Heidelberg (2006). Scholar
  23. 23.
    Marx, D., Pilipczuk, M.: Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask). In: Proceedings of the 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), pp. 542 (2014)Google Scholar
  24. 24.
    Matoušek, J., Thomas, R.: On the complexity of finding ISO-and other morphisms for partial k-trees. Discret. Math. 108(1), 343–364 (1992)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Nešetřil, J.: Amalgamation of graphs and its applications. Ann. New York Acad. Sci. 319(1), 415–428 (1979)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Robertson, N., Seymour, P.D.: Graph minors. XVI. Excluding a non-planar graph. J. Combin. Theory, Ser. B 89(1), 43–76 (2003)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Seymour, P.D., Weaver, R.: A generalization of chordal graphs. J. Graph Theory 8(2), 241–251 (1984)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Yang, Y., Chen, Y.: The thickness of amalgamations and cartesian product of graphs. Discuss. Math. Graph Theory 37(3), 561–572 (2017)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.University of BergenBergenNorway

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