220 research outputs found
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
A degree and forbidden subgraph condition for a k-contractible edge
An edge in a k-conected graph is said to be k-contractible if the contraction of it results in a k-connected graph. We say that k-connected graph G satis es “ degree-sum conditon ”if Σx2V (W)degG(x) 3k +2 holds for any connected subgraph W of G with |W|= 3. Let k be an integer such that k 5. We prove that if a k-connected graph with no K1+C4 satis es degree-sum condition, then it has a k-contractible edge.電気通信大学201
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.
On \pi-surfaces of four-dimensional parallelohedra
We show that every four-dimensional parallelohedron P satisfies a recently
found condition of Garber, Gavrilyuk & Magazinov sufficient for the Voronoi
conjecture being true for P. Namely we show that for every four-dimensional
parallelohedron P the group of rational first homologies of its \pi-surface is
generated by half-belt cycles.Comment: 16 pages, 7 figure
Matching Augmentation via Simultaneous Contractions
We consider the matching augmentation problem (MAP), where a matching of a graph needs to be extended into a 2-edge-connected spanning subgraph by adding the minimum number of edges to it. We present a polynomial-time algorithm with an approximation ratio of 13/8 = 1.625 improving upon an earlier 5/3-approximation. The improvement builds on a new ?-approximation preserving reduction for any ? ? 3/2 from arbitrary MAP instances to well-structured instances that do not contain certain forbidden structures like parallel edges, small separators, and contractible subgraphs. We further introduce, as key ingredients, the technique of repeated simultaneous contractions and provide improved lower bounds for instances that cannot be contracted
Algebraic properties of toric rings of graphs
Let be a simple graph. We investigate the Cohen-Macaulayness and
algebraic invariants, such as the Castelnuovo-Mumford regularity and the
projective dimension, of the toric ring via those of toric rings
associated to induced subgraphs of .Comment: 18 pages; changed title and re-organized sections to better exhibit
results; correct the last main theore
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