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Graph Amalgamation Under Logical Constraints

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Graph-Theoretic Concepts in Computer Science (WG 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11159))

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Abstract

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.

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Notes

  1. 1.

    Graph morphisms will be properly defined in Sect. 2.

  2. 2.

    Definability of these polynomials require access to a total ordering of the edge set of the graph. On graphs of constant treewidth such orderings are by themselves CMSO definable.

  3. 3.

    We note that in texts dealing with similar notions of decomposition, it is customary to define a bag of width t as a graph with at most t vertices together with a function that labels the vertices of these graphs with numbers from \(\{1,...,t\}\). Our notion of t-concrete bag, on the other hand, may be regarded as a representation of a graph with at most t vertices injectively labeled with numbers from \(\{1,...t\}\) and at most one edge. Within this point of view, the representation used here is a syntactic restriction of the former. On the other hand, any decomposition which uses bags with arbitrary graphs of size t can be converted into a t-concrete decomposition, by expanding each bag into a sequence of \(t^2\) concrete bags.

  4. 4.

    Both Planarity and Hamiltonicity are CMSO-definable (note that our definition of CMSO logic allows edge-set quantifications).

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Acknowledgements

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

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Authors and Affiliations

  1. University of Bergen, Postboks 7803, 5020, Bergen, Norway

    Mateus de Oliveira Oliveira

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  1. Mateus de Oliveira Oliveira
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Correspondence to Mateus de Oliveira Oliveira .

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Editors and Affiliations

  1. Universität Rostock, Rostock, Germany

    Andreas Brandstädt

  2. Brandenburgische Technische Universität Cottbus-Senftenberg, Cottbus, Germany

    Ekkehard Köhler

  3. Brandenburgische Technische Universität Cottbus-Senftenberg, Cottbus, Germany

    Klaus Meer

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de Oliveira Oliveira, M. (2018). Graph Amalgamation Under Logical Constraints. In: Brandstädt, A., Köhler, E., Meer, K. (eds) Graph-Theoretic Concepts in Computer Science. WG 2018. Lecture Notes in Computer Science(), vol 11159. Springer, Cham. https://doi.org/10.1007/978-3-030-00256-5_13

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  • DOI: https://doi.org/10.1007/978-3-030-00256-5_13

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