5,475 research outputs found
Hypercellular graphs: partial cubes without as partial cube minor
We investigate the structure of isometric subgraphs of hypercubes (i.e.,
partial cubes) which do not contain finite convex subgraphs contractible to the
3-cube minus one vertex (here contraction means contracting the edges
corresponding to the same coordinate of the hypercube). Extending similar
results for median and cellular graphs, we show that the convex hull of an
isometric cycle of such a graph is gated and isomorphic to the Cartesian
product of edges and even cycles. Furthermore, we show that our graphs are
exactly the class of partial cubes in which any finite convex subgraph can be
obtained from the Cartesian products of edges and even cycles via successive
gated amalgams. This decomposition result enables us to establish a variety of
results. In particular, it yields that our class of graphs generalizes median
and cellular graphs, which motivates naming our graphs hypercellular.
Furthermore, we show that hypercellular graphs are tope graphs of zonotopal
complexes of oriented matroids. Finally, we characterize hypercellular graphs
as being median-cell -- a property naturally generalizing the notion of median
graphs.Comment: 35 pages, 6 figures, added example answering Question 1 from earlier
draft (Figure 6.
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
Bucolic Complexes
We introduce and investigate bucolic complexes, a common generalization of
systolic complexes and of CAT(0) cubical complexes. They are defined as simply
connected prism complexes satisfying some local combinatorial conditions. We
study various approaches to bucolic complexes: from graph-theoretic and
topological perspective, as well as from the point of view of geometric group
theory. In particular, we characterize bucolic complexes by some properties of
their 2-skeleta and 1-skeleta (that we call bucolic graphs), by which several
known results are generalized. We also show that locally-finite bucolic
complexes are contractible, and satisfy some nonpositive-curvature-like
properties.Comment: 45 pages, 4 figure
Convexities related to path properties on graphs; a unified approach
Path properties, such as 'geodesic', 'induced', 'all paths' define a convexity on a connected graph. The general notion of path property, introduced in this paper, gives rise to a comprehensive survey of results obtained by different authors for a variety of path properties, together with a number of new results. We pay special attention to convexities defined by path properties on graph products and the classical convexity invariants, such as the Caratheodory, Helly and Radon numbers in relation with graph invariants, such as clique numbers and other graph properties.
Ramified rectilinear polygons: coordinatization by dendrons
Simple rectilinear polygons (i.e. rectilinear polygons without holes or
cutpoints) can be regarded as finite rectangular cell complexes coordinatized
by two finite dendrons. The intrinsic -metric is thus inherited from the
product of the two finite dendrons via an isometric embedding. The rectangular
cell complexes that share this same embedding property are called ramified
rectilinear polygons. The links of vertices in these cell complexes may be
arbitrary bipartite graphs, in contrast to simple rectilinear polygons where
the links of points are either 4-cycles or paths of length at most 3. Ramified
rectilinear polygons are particular instances of rectangular complexes obtained
from cube-free median graphs, or equivalently simply connected rectangular
complexes with triangle-free links. The underlying graphs of finite ramified
rectilinear polygons can be recognized among graphs in linear time by a
Lexicographic Breadth-First-Search. Whereas the symmetry of a simple
rectilinear polygon is very restricted (with automorphism group being a
subgroup of the dihedral group ), ramified rectilinear polygons are
universal: every finite group is the automorphism group of some ramified
rectilinear polygon.Comment: 27 pages, 6 figure
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