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Robust self-assembly of graphs

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Abstract

Self-assembly is a process in which small building blocks interact autonomously to form larger structures. A recently studied model of self-assembly is the Accretive Graph Assembly Model whereby an edge-weighted graph is assembled one vertex at a time starting from a designated seed vertex. The weight of an edge specifies the magnitude of attraction (positive weight) or repulsion (negative weight) between adjacent vertices. It is feasible to add a vertex to the assembly if the total attraction minus repulsion of the already built neighbors exceeds a certain threshold, called the assembly temperature. This model naturally generalizes the extensively studied Tile Assembly Model. A natural question in graph self-assembly is to determine whether or not there exists a sequence of feasible vertex additions to realize the entire graph. However, even when it is feasible to realize the assembly, not much can be inferred about its likelihood of realization in practice due to the uncontrolled nature of the self-assembly process. Motivated by this, we introduce the robust self-assembly problem where the goal is to determine if every possible sequence of feasible vertex additions leads to the completion of the assembly. We show that the robust self-assembly problem is co-NP-complete even on planar graphs with two distinct edge weights. We then examine the tractability of the robust self-assembly problem on a natural subclass of planar graphs, namely grid graphs. We identify structural conditions that determine whether or not a grid graph can be robustly self-assembled, and give poly-time algorithms to determine this for several interesting cases of the problem. Finally, we also show that the problem of counting the number of feasible orderings that lead to the completion of an assembly is #P-complete.

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Authors and Affiliations

  1. Google, Inc., New York, NY, 10011, USA

    Stanislav Angelov

  2. Department of Computer and Information Science, School of Engineering and Applied Sciences, University of Pennsylvania, Philadelphia, PA, 19104, USA

    Sanjeev Khanna

  3. Department of Mathematics, School of Arts and Sciences, University of Pennsylvania, Philadelphia, PA, 19104, USA

    Mirkó Visontai

Authors
  1. Stanislav Angelov
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  2. Sanjeev Khanna
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  3. Mirkó Visontai
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Corresponding author

Correspondence to Stanislav Angelov.

Additional information

A significant portion of this work was done while the first author was at the Department of Computer and Information Science, School of Engineering and Applied Sciences, University of Pennsylvania.

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Angelov, S., Khanna, S. & Visontai, M. Robust self-assembly of graphs. Nat Comput 9, 111–133 (2010). https://doi.org/10.1007/s11047-009-9149-5

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  • DOI: https://doi.org/10.1007/s11047-009-9149-5

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