19,818 research outputs found
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
Strongly Monotone Drawings of Planar Graphs
A straight-line drawing of a graph is a monotone drawing if for each pair of
vertices there is a path which is monotonically increasing in some direction,
and it is called a strongly monotone drawing if the direction of monotonicity
is given by the direction of the line segment connecting the two vertices.
We present algorithms to compute crossing-free strongly monotone drawings for
some classes of planar graphs; namely, 3-connected planar graphs, outerplanar
graphs, and 2-trees. The drawings of 3-connected planar graphs are based on
primal-dual circle packings. Our drawings of outerplanar graphs are based on a
new algorithm that constructs strongly monotone drawings of trees which are
also convex. For irreducible trees, these drawings are strictly convex
Asymptotic expansion of the multi-orientable random tensor model
Three-dimensional random tensor models are a natural generalization of the
celebrated matrix models. The associated tensor graphs, or 3D maps, can be
classified with respect to a particular integer or half-integer, the degree of
the respective graph. In this paper we analyze the general term of the
asymptotic expansion in N, the size of the tensor, of a particular random
tensor model, the multi-orientable tensor model. We perform their enumeration
and we establish which are the dominant configurations of a given degree.Comment: 27 pages, 24 figures, several minor modifications have been made, one
figure has been added; accepted for publication in "Electronic Journal of
Combinatorics
Unit Grid Intersection Graphs: Recognition and Properties
It has been known since 1991 that the problem of recognizing grid
intersection graphs is NP-complete. Here we use a modified argument of the
above result to show that even if we restrict to the class of unit grid
intersection graphs (UGIGs), the recognition remains hard, as well as for all
graph classes contained inbetween. The result holds even when considering only
graphs with arbitrarily large girth. Furthermore, we ask the question of
representing UGIGs on grids of minimal size. We show that the UGIGs that can be
represented in a square of side length 1+epsilon, for a positive epsilon no
greater than 1, are exactly the orthogonal ray graphs, and that there exist
families of trees that need an arbitrarily large grid
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