446 research outputs found
Countable locally 2-arc-transitive bipartite graphs
We present an order-theoretic approach to the study of countably infinite
locally 2-arc-transitive bipartite graphs. Our approach is motivated by
techniques developed by Warren and others during the study of cycle-free
partial orders. We give several new families of previously unknown countably
infinite locally-2-arc-transitive graphs, each family containing continuum many
members. These examples are obtained by gluing together copies of incidence
graphs of semilinear spaces, satisfying a certain symmetry property, in a
tree-like way. In one case we show how the classification problem for that
family relates to the problem of determining a certain family of highly
arc-transitive digraphs. Numerous illustrative examples are given.Comment: 29 page
Strong Connectivity in Directed Graphs under Failures, with Application
In this paper, we investigate some basic connectivity problems in directed
graphs (digraphs). Let be a digraph with edges and vertices, and
let be the digraph obtained after deleting edge from . As
a first result, we show how to compute in worst-case time: The
total number of strongly connected components in , for all edges
in . The size of the largest and of the smallest strongly
connected components in , for all edges in .
Let be strongly connected. We say that edge separates two vertices
and , if and are no longer strongly connected in .
As a second set of results, we show how to build in time -space
data structures that can answer in optimal time the following basic
connectivity queries on digraphs: Report in worst-case time all
the strongly connected components of , for a query edge .
Test whether an edge separates two query vertices in worst-case
time. Report all edges that separate two query vertices in optimal
worst-case time, i.e., in time , where is the number of separating
edges. (For , the time is ).
All of the above results extend to vertex failures. All our bounds are tight
and are obtained with a common algorithmic framework, based on a novel compact
representation of the decompositions induced by the -connectivity (i.e.,
-edge and -vertex) cuts in digraphs, which might be of independent
interest. With the help of our data structures we can design efficient
algorithms for several other connectivity problems on digraphs and we can also
obtain in linear time a strongly connected spanning subgraph of with
edges that maintains the -connectivity cuts of and the decompositions
induced by those cuts.Comment: An extended abstract of this work appeared in the SODA 201
The Order Dimension of the Poset of Regions in a Hyperplane Arrangement
We show that the order dimension of the weak order on a Coxeter group of type
A, B or D is equal to the rank of the Coxeter group, and give bounds on the
order dimensions for the other finite types. This result arises from a unified
approach which, in particular, leads to a simpler treatment of the previously
known cases, types A and B. The result for weak orders follows from an upper
bound on the dimension of the poset of regions of an arbitrary hyperplane
arrangement. In some cases, including the weak orders, the upper bound is the
chromatic number of a certain graph. For the weak orders, this graph has the
positive roots as its vertex set, and the edges are related to the pairwise
inner products of the roots.Comment: Minor changes, including a correction and an added figure in the
proof of Proposition 2.2. 19 pages, 6 figure
Decremental Single-Source Reachability in Planar Digraphs
In this paper we show a new algorithm for the decremental single-source
reachability problem in directed planar graphs. It processes any sequence of
edge deletions in total time and explicitly
maintains the set of vertices reachable from a fixed source vertex. Hence, if
all edges are eventually deleted, the amortized time of processing each edge
deletion is only , which improves upon a previously
known solution. We also show an algorithm for decremental
maintenance of strongly connected components in directed planar graphs with the
same total update time. These results constitute the first almost optimal (up
to polylogarithmic factors) algorithms for both problems.
To the best of our knowledge, these are the first dynamic algorithms with
polylogarithmic update times on general directed planar graphs for non-trivial
reachability-type problems, for which only polynomial bounds are known in
general graphs
Structural Routability of n-Pairs Information Networks
Information does not generally behave like a conservative fluid flow in
communication networks with multiple sources and sinks. However, it is often
conceptually and practically useful to be able to associate separate data
streams with each source-sink pair, with only routing and no coding performed
at the network nodes. This raises the question of whether there is a nontrivial
class of network topologies for which achievability is always equivalent to
routability, for any combination of source signals and positive channel
capacities. This chapter considers possibly cyclic, directed, errorless
networks with n source-sink pairs and mutually independent source signals. The
concept of downward dominance is introduced and it is shown that, if the
network topology is downward dominated, then the achievability of a given
combination of source signals and channel capacities implies the existence of a
feasible multicommodity flow.Comment: The final publication is available at link.springer.com
http://link.springer.com/chapter/10.1007/978-3-319-02150-8_
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