1,132 research outputs found
A note on forbidding clique immersions
Robertson and Seymour proved that the relation of graph immersion is
well-quasi-ordered for finite graphs. Their proof uses the results of graph
minors theory. Surprisingly, there is a very short proof of the corresponding
rough structure theorem for graphs without -immersions; it is based on the
Gomory-Hu theorem. The same proof also works to establish a rough structure
theorem for Eulerian digraphs without -immersions, where
denotes the bidirected complete digraph of order
Nondeterministic graph property testing
A property of finite graphs is called nondeterministically testable if it has
a "certificate" such that once the certificate is specified, its correctness
can be verified by random local testing. In this paper we study certificates
that consist of one or more unary and/or binary relations on the nodes, in the
case of dense graphs. Using the theory of graph limits, we prove that
nondeterministically testable properties are also deterministically testable.Comment: Version 2: 11 pages; we allow orientation in the certificate,
describe new application
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
Single Source - All Sinks Max Flows in Planar Digraphs
Let G = (V,E) be a planar n-vertex digraph. Consider the problem of computing
max st-flow values in G from a fixed source s to all sinks t in V\{s}. We show
how to solve this problem in near-linear O(n log^3 n) time. Previously, no
better solution was known than running a single-source single-sink max flow
algorithm n-1 times, giving a total time bound of O(n^2 log n) with the
algorithm of Borradaile and Klein.
An important implication is that all-pairs max st-flow values in G can be
computed in near-quadratic time. This is close to optimal as the output size is
Theta(n^2). We give a quadratic lower bound on the number of distinct max flow
values and an Omega(n^3) lower bound for the total size of all min cut-sets.
This distinguishes the problem from the undirected case where the number of
distinct max flow values is O(n).
Previous to our result, no algorithm which could solve the all-pairs max flow
values problem faster than the time of Theta(n^2) max-flow computations for
every planar digraph was known.
This result is accompanied with a data structure that reports min cut-sets.
For fixed s and all t, after O(n^{3/2} log^{3/2} n) preprocessing time, it can
report the set of arcs C crossing a min st-cut in time roughly proportional to
the size of C.Comment: 25 pages, 4 figures; extended abstract appeared in FOCS 201
Sink-Stable Sets of Digraphs
We introduce the notion of sink-stable sets of a digraph and prove a min-max
formula for the maximum cardinality of the union of k sink-stable sets. The
results imply a recent min-max theorem of Abeledo and Atkinson on the Clar
number of bipartite plane graphs and a sharpening of Minty's coloring theorem.
We also exhibit a link to min-max results of Bessy and Thomasse and of Sebo on
cyclic stable sets
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