342 research outputs found
On the Complexity of Role Colouring Planar Graphs, Trees and Cographs
We prove several results about the complexity of the role colouring problem.
A role colouring of a graph is an assignment of colours to the vertices of
such that two vertices of the same colour have identical sets of colours in
their neighbourhoods. We show that the problem of finding a role colouring with
colours is NP-hard for planar graphs. We show that restricting the
problem to trees yields a polynomially solvable case, as long as is either
constant or has a constant difference with , the number of vertices in the
tree. Finally, we prove that cographs are always -role-colourable for
and construct such a colouring in polynomial time
Phase Transitions in Edge-Weighted Exponential Random Graphs: Near-Degeneracy and Universality
Conventionally used exponential random graphs cannot directly model weighted
networks as the underlying probability space consists of simple graphs only.
Since many substantively important networks are weighted, this limitation is
especially problematic. We extend the existing exponential framework by
proposing a generic common distribution for the edge weights. Minimal
assumptions are placed on the distribution, that is, it is non-degenerate and
supported on the unit interval. By doing so, we recognize the essential
properties associated with near-degeneracy and universality in edge-weighted
exponential random graphs.Comment: 15 pages, 4 figures. This article extends arXiv:1607.04084, which
derives general formulas for the normalization constant and characterizes
phase transitions in exponential random graphs with uniformly distributed
edge weights. The present article places minimal assumptions on the
edge-weight distribution, thereby recognizing essential properties associated
with near-degeneracy and universalit
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