151 research outputs found
Computational complexity of reconstruction and isomorphism testing for designs and line graphs
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 - designs. For this class of
highly regular graphs, we obtain a worst-case running time of for bounded parameters . In a first step, our approach
makes use of the Babai--Luks algorithm to compute canonical forms of
-designs. In a second step, we show that -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
Graph homology: Koszul and Verdier duality
We show that Verdier duality for certain sheaves on the moduli spaces of
graphs associated to Koszul operads corresponds to Koszul duality of operads.
This in particular gives a conceptual explanation of the appearance of graph
cohomology of both the commutative and Lie types in computations of the
cohomology of the outer automorphism group of a free group. Another consequence
is an explicit computation of dualizing sheaves on spaces of metric graphs,
thus characterizing to which extent these spaces are different from oriented
orbifolds. We also provide a relation between the cohomology of the space of
metric ribbon graphs, known to be homotopy equivalent to the moduli space of
Riemann surfaces, and the cohomology of a certain sheaf on the space of usual
metric graphs.Comment: 13 page
Tropical curves, graph complexes, and top weight cohomology of M_g
We study the topology of a space parametrizing stable tropical curves of
genus g with volume 1, showing that its reduced rational homology is
canonically identified with both the top weight cohomology of M_g and also with
the genus g part of the homology of Kontsevich's graph complex. Using a theorem
of Willwacher relating this graph complex to the Grothendieck-Teichmueller Lie
algebra, we deduce that H^{4g-6}(M_g;Q) is nonzero for g=3, g=5, and g at least
7. This disproves a recent conjecture of Church, Farb, and Putman as well as an
older, more general conjecture of Kontsevich. We also give an independent proof
of another theorem of Willwacher, that homology of the graph complex vanishes
in negative degrees.Comment: 31 pages. v2: streamlined exposition. Final version, to appear in J.
Amer. Math. So
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