4,968 research outputs found

    Computational complexity of reconstruction and isomorphism testing for designs and line graphs

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    Graphs with high symmetry or regularity are the main source for experimentally hard instances of the notoriously difficult graph isomorphism problem. In this paper, we study the computational complexity of isomorphism testing for line graphs of tt-(v,k,λ)(v,k,\lambda) designs. For this class of highly regular graphs, we obtain a worst-case running time of O(vlogv+O(1))O(v^{\log v + O(1)}) for bounded parameters t,k,λt,k,\lambda. In a first step, our approach makes use of the Babai--Luks algorithm to compute canonical forms of tt-designs. In a second step, we show that tt-designs can be reconstructed from their line graphs in polynomial-time. The first is algebraic in nature, the second purely combinatorial. For both, profound structural knowledge in design theory is required. Our results extend earlier complexity results about isomorphism testing of graphs generated from Steiner triple systems and block designs.Comment: 12 pages; to appear in: "Journal of Combinatorial Theory, Series A

    Collective decision-making on triadic graphs

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    Many real-world networks exhibit community structures and non-trivial clustering associated with the occurrence of a considerable number of triangular subgraphs known as triadic motifs. Triads are a set of distinct triangles that do not share an edge with any other triangle in the network. Network motifs are subgraphs that occur significantly more often compared to random topologies. Two prominent examples, the feedforward loop and the feedback loop, occur in various real-world networks such as gene-regulatory networks, food webs or neuronal networks. However, as triangular connections are also prevalent in communication topologies of complex collective systems, it is worthwhile investigating the influence of triadic motifs on the collective decision-making dynamics. To this end, we generate networks called Triadic Graphs (TGs) exclusively from distinct triadic motifs. We then apply TGs as underlying topologies of systems with collective dynamics inspired from locust marching bands. We demonstrate that the motif type constituting the networks can have a paramount influence on group decision-making that cannot be explained solely in terms of the degree distribution. We find that, in contrast to the feedback loop, when the feedforward loop is the dominant subgraph, the resulting network is hierarchical and inhibits coherent behavior

    Regular Two-graphs from the Even Unimodular Lattice E8⊕E8

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    AbstractStarting from the even unimodular latticeE8⊕E8,one constructs odd systems (i.e. sets of vectors with odd inner products) of 546 vectors using results of Deza and Grishukhin. One studies the subsystems consisting of 36 pairs of opposite vectors spanning equiangular lines. These subsystems represent regular two-graphs. This gives 100 such two-graphs and they coincide with the first 100 in a list of 227 two-graphs generated by E. Spence. Using the root systems of the sublattices generated by the 100 odd systems, the set of the 100 two-graphs is divided into seven classes. The first four classes correspond to the 23 Steiner triple system on 15 points containing a head, i.e. a Fano plane
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