888 research outputs found
Colored Point-Set Embeddings of Acyclic Graphs.
We show that any planar drawing of a forest of three stars whose vertices are
constrained to be at fixed vertex locations may require
edges each having bends in the worst case. The lower
bound holds even when the function that maps vertices to points is not a
bijection but it is defined by a 3-coloring. In contrast, a constant number of
bends per edge can be obtained for 3-colored paths and for 3-colored
caterpillars whose leaves all have the same color. Such results answer to a
long standing open problem.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Small Superpatterns for Dominance Drawing
We exploit the connection between dominance drawings of directed acyclic
graphs and permutations, in both directions, to provide improved bounds on the
size of universal point sets for certain types of dominance drawing and on
superpatterns for certain natural classes of permutations. In particular we
show that there exist universal point sets for dominance drawings of the Hasse
diagrams of width-two partial orders of size O(n^{3/2}), universal point sets
for dominance drawings of st-outerplanar graphs of size O(n\log n), and
universal point sets for dominance drawings of directed trees of size O(n^2).
We show that 321-avoiding permutations have superpatterns of size O(n^{3/2}),
riffle permutations (321-, 2143-, and 2413-avoiding permutations) have
superpatterns of size O(n), and the concatenations of sequences of riffles and
their inverses have superpatterns of size O(n\log n). Our analysis includes a
calculation of the leading constants in these bounds.Comment: ANALCO 2014, This version fixes an error in the leading constant of
the 321-superpattern siz
Formality of a higher-codimensional Swiss-Cheese operad
We study configurations of points in the complement of a linear subspace
inside a Euclidean space, with . We define a higher-codimensional Swiss-Cheese operad
associated to such configurations, a variant of the
classical Swiss-Cheese operad. The operad is weakly
equivalent to the operad of locally constant factorization algebras on the
stratified space . We prove that
this operad is formal over .Comment: 50 pages, comments welcome. v2: Added two appendices and corrected
Section 5.
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