14,416 research outputs found
Complexity dichotomy on partial grid recognition
Deciding whether a graph can be embedded in a grid using only unit-length
edges is NP-complete, even when restricted to binary trees. However, it is not
difficult to devise a number of graph classes for which the problem is
polynomial, even trivial. A natural step, outstanding thus far, was to provide
a broad classification of graphs that make for polynomial or NP-complete
instances. We provide such a classification based on the set of allowed vertex
degrees in the input graphs, yielding a full dichotomy on the complexity of the
problem. As byproducts, the previous NP-completeness result for binary trees
was strengthened to strictly binary trees, and the three-dimensional version of
the problem was for the first time proven to be NP-complete. Our results were
made possible by introducing the concepts of consistent orientations and robust
gadgets, and by showing how the former allows NP-completeness proofs by local
replacement even in the absence of the latter
Applications of Structural Balance in Signed Social Networks
We present measures, models and link prediction algorithms based on the
structural balance in signed social networks. Certain social networks contain,
in addition to the usual 'friend' links, 'enemy' links. These networks are
called signed social networks. A classical and major concept for signed social
networks is that of structural balance, i.e., the tendency of triangles to be
'balanced' towards including an even number of negative edges, such as
friend-friend-friend and friend-enemy-enemy triangles. In this article, we
introduce several new signed network analysis methods that exploit structural
balance for measuring partial balance, for finding communities of people based
on balance, for drawing signed social networks, and for solving the problem of
link prediction. Notably, the introduced methods are based on the signed graph
Laplacian and on the concept of signed resistance distances. We evaluate our
methods on a collection of four signed social network datasets.Comment: 37 page
Combinatorics in the Art of the Twentieth Century
This paper is motivated by a question I asked myself: How can combinatorial structures be used in a work of art? Immediately, other questions arose: Whether there are artists that work or think combinatorially? If so, what works have they produced in this way? What are the similarities and differences between art works produced using
combinatorics? This paper presents the first results of the attempt to answer these questions, being a survey of a selection of works that use or contain combinatorics in some way, including music, literature and visual arts, focusing on the twentieth century.Postprint (published version
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