33 research outputs found
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
On the connectivity and restricted edge-connectivity of 3-arc graphs
A 3âarc of a graph G is a 4-tuple (y, a, b, x) of vertices such that both (y, a, b) and (a, b, x) are paths of length two in G. Let ââG denote the symmetric digraph of a graph G. The 3-arc graph X(G) of a given graph G is defined to have vertices the arcs of ââG. Two vertices (ay), (bx) are adjacent in X(G) if and only if (y, a, b, x) is a 3-arc of G. The purpose of this work is to study the edge-connectivity and restricted edge-connectivity of 3-arc graphs.We prove that the 3-arc graph X(G) of every connected graph G of minimum degree ÎŽ(G) â„ 3 has edge-connectivity λ(X(G)) â„ (ÎŽ(G) â 1)2; and restricted edge- connectivity λ(2)(X(G)) â„ 2(ÎŽ(G) â 1)2 â 2 if Îș(G) â„ 2. We also provide examples showing that all these bounds are sharp.Peer Reviewe
Recommended from our members
Robust Detection of Dynamical Change in Scalp EEG
We present a robust, model-independent technique for measuring changes in the dynamics underlying nonlinear time-serial data. We define indicators of dynamical change by comparing distribution functions on the attractor via L{sub 1}-distance and X{sup 2} statistics. We apply the measures to scalp EEG data with the objective of capturing the transition between non-seizure and epileptic brain activity in a timely, accurate, and non-invasive manner. We find a clear superiority of the new metrics in comparison to traditional nonlinear measures as discriminators of dynamical change
Cover Time and Broadcast Time
We introduce a new technique for bounding the cover time of random walks by relating it to the runtime of randomized broadcast. In particular, we strongly confirm for dense graphs the intuition of Chandra et al. (1997) that ``the cover time of the graph is an appropriate metric for the performance of certain kinds of randomized broadcast algorithms\u27\u27. In more detail, our results are as follows:
begin{itemize}
item For any graph of size and minimum degree , we have , where denotes the quotient of the cover time and broadcast time. This bound is tight for binary trees and tight up to logarithmic factors for many graphs including hypercubes, expanders and lollipop graphs.
item For any -regular (or almost -regular) graph it holds that . Together with our upper bound on , this lower bound strongly confirms the intuition of Chandra et al.~for graphs with minimum degree , since then the cover time equals the broadcast time multiplied by (neglecting logarithmic factors).
item Conversely, for any we construct almost -regular graphs that satisfy . Since any regular expander satisfies , the strong relationship given above does not hold if is polynomially smaller than .
end{itemize}
Our bounds also demonstrate that the relationship between cover time and broadcast time is much stronger than the known relationships between any of them and the mixing time (or the closely related spectral gap)