30,823 research outputs found
Pixel and Voxel Representations of Graphs
We study contact representations for graphs, which we call pixel
representations in 2D and voxel representations in 3D. Our representations are
based on the unit square grid whose cells we call pixels in 2D and voxels in
3D. Two pixels are adjacent if they share an edge, two voxels if they share a
face. We call a connected set of pixels or voxels a blob. Given a graph, we
represent its vertices by disjoint blobs such that two blobs contain adjacent
pixels or voxels if and only if the corresponding vertices are adjacent. We are
interested in the size of a representation, which is the number of pixels or
voxels it consists of.
We first show that finding minimum-size representations is NP-complete. Then,
we bound representation sizes needed for certain graph classes. In 2D, we show
that, for -outerplanar graphs with vertices, pixels are
always sufficient and sometimes necessary. In particular, outerplanar graphs
can be represented with a linear number of pixels, whereas general planar
graphs sometimes need a quadratic number. In 3D, voxels are
always sufficient and sometimes necessary for any -vertex graph. We improve
this bound to for graphs of treewidth and to
for graphs of genus . In particular, planar graphs
admit representations with voxels
Visibility Representations of Boxes in 2.5 Dimensions
We initiate the study of 2.5D box visibility representations (2.5D-BR) where
vertices are mapped to 3D boxes having the bottom face in the plane and
edges are unobstructed lines of sight parallel to the - or -axis. We
prove that: Every complete bipartite graph admits a 2.5D-BR; The
complete graph admits a 2.5D-BR if and only if ; Every
graph with pathwidth at most admits a 2.5D-BR, which can be computed in
linear time. We then turn our attention to 2.5D grid box representations
(2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit
square at integer coordinates. We show that an -vertex graph that admits a
2.5D-GBR has at most edges and this bound is tight. Finally,
we prove that deciding whether a given graph admits a 2.5D-GBR with a given
footprint is NP-complete. The footprint of a 2.5D-BR is the set of
bottom faces of the boxes in .Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Situating graphs as workplace knowledge
We investigate the use and knowledge of graphs in the context of a large industrial factory. We are particularly interested in the question of "transparency", a question that has been extensively considered in the general literature on tool use, and more recently, by Michael Roth and his colleagues in the context of scientific work. Roth uses the notion of transparency to characterise instances of graph use by highly educated scientists in cases where the context was familiar: the scientists were able to read the situation "through" the graph. This paper explores the limits of the validity of the transparency metaphor. We present two vignettes of actual graph use by a factory worker, and contrast his actions and knowledge with that of a highly-qualified process engineer working on the same production line. We note that in neither case were the graphs transparent. We argue that a fuller account that describes a spectrum of transparency is needed, and we seek to achieve this by adopting some elements of a semiotic approach that enhance a strictly activity-theoretical view
Crystal bases for quantum affine algebras and combinatorics of Young walls
In this paper, we give a realization of crystal bases for quantum affine
algebras using some new combinatorial objects which we call the Young walls.
The Young walls consist of colored blocks with various shapes that are built on
the given ground-state wall and can be viewed as generalizations of Young
diagrams. The rules for building Young walls and the action of Kashiwara
operators are given explicitly in terms of combinatorics of Young walls. The
crystal graphs for basic representations are characterized as the set of all
reduced proper Young walls. The characters of basic representations can be
computed easily by counting the number of colored blocks that have been added
to the ground-state wall
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