2,082 research outputs found
The Widths of Strict Outerconfluent Graphs
Strict outerconfluent drawing is a style of graph drawing in which vertices
are drawn on the boundary of a disk, adjacencies are indicated by the existence
of smooth curves through a system of tracks within the disk, and no two
adjacent vertices are connected by more than one of these smooth tracks. We
investigate graph width parameters on the graphs that have drawings in this
style. We prove that the clique-width of these graphs is unbounded, but their
twin-width is bounded.Comment: 15 pages, 2 figure
Normalization for planar string diagrams and a quadratic equivalence algorithm
In the graphical calculus of planar string diagrams, equality is generated by
exchange moves, which swap the heights of adjacent vertices. We show that left-
and right-handed exchanges each give strongly normalizing rewrite strategies
for connected string diagrams. We use this result to give a linear-time
solution to the equivalence problem in the connected case, and a quadratic
solution in the general case. We also give a stronger proof of the Joyal-Street
coherence theorem, settling Selinger's conjecture on recumbent isotopy
Cyclic Datatypes modulo Bisimulation based on Second-Order Algebraic Theories
Cyclic data structures, such as cyclic lists, in functional programming are
tricky to handle because of their cyclicity. This paper presents an
investigation of categorical, algebraic, and computational foundations of
cyclic datatypes. Our framework of cyclic datatypes is based on second-order
algebraic theories of Fiore et al., which give a uniform setting for syntax,
types, and computation rules for describing and reasoning about cyclic
datatypes. We extract the "fold" computation rules from the categorical
semantics based on iteration categories of Bloom and Esik. Thereby, the rules
are correct by construction. We prove strong normalisation using the General
Schema criterion for second-order computation rules. Rather than the fixed
point law, we particularly choose Bekic law for computation, which is a key to
obtaining strong normalisation. We also prove the property of "Church-Rosser
modulo bisimulation" for the computation rules. Combining these results, we
have a remarkable decidability result of the equational theory of cyclic data
and fold.Comment: 38 page
Edge-Path Bundling: A Less Ambiguous Edge Bundling Approach
Edge bundling techniques cluster edges with similar attributes (i.e. similarity in direction and proximity) together to reduce the visual clutter. All edge bundling techniques to date implicitly or explicitly cluster groups of individual edges, or parts of them, together based on these attributes. These clusters can result in ambiguous connections that do not exist in the data. Confluent drawings of networks do not have these ambiguities, but require the layout to be computed as part of the bundling process. We devise a new bundling method, Edge-Path bundling, to simplify edge clutter while greatly reducing ambiguities compared to previous bundling techniques. Edge-Path bundling takes a layout as input and clusters each edge along a weighted, shortest path to limit its deviation from a straight line. Edge-Path bundling does not incur independent edge ambiguities typically seen in all edge bundling methods, and the level of bundling can be tuned through shortest path distances, Euclidean distances, and combinations of the two. Also, directed edge bundling naturally emerges from the model. Through metric evaluations, we demonstrate the advantages of Edge-Path bundling over other techniques
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