47,269 research outputs found
Structurally Parameterized d-Scattered Set
In -Scattered Set we are given an (edge-weighted) graph and are asked to
select at least vertices, so that the distance between any pair is at least
, thus generalizing Independent Set. We provide upper and lower bounds on
the complexity of this problem with respect to various standard graph
parameters. In particular, we show the following:
- For any , an -time algorithm, where
is the treewidth of the input graph.
- A tight SETH-based lower bound matching this algorithm's performance. These
generalize known results for Independent Set.
- -Scattered Set is W[1]-hard parameterized by vertex cover (for
edge-weighted graphs), or feedback vertex set (for unweighted graphs), even if
is an additional parameter.
- A single-exponential algorithm parameterized by vertex cover for unweighted
graphs, complementing the above-mentioned hardness.
- A -time algorithm parameterized by tree-depth
(), as well as a matching ETH-based lower bound, both for
unweighted graphs.
We complement these mostly negative results by providing an FPT approximation
scheme parameterized by treewidth. In particular, we give an algorithm which,
for any error parameter , runs in time
and returns a
-scattered set of size , if a -scattered set of the same
size exists
The skew energy of random oriented graphs
Given a graph , let be an oriented graph of with the
orientation and skew-adjacency matrix . The skew energy
of the oriented graph , denoted by , is
defined as the sum of the absolute values of all the eigenvalues of
. In this paper, we study the skew energy of random oriented
graphs and formulate an exact estimate of the skew energy for almost all
oriented graphs by generalizing Wigner's semicircle law. Moreover, we consider
the skew energy of random regular oriented graphs , and get an
exact estimate of the skew energy for almost all regular oriented graphs.Comment: 12 pages. arXiv admin note: text overlap with arXiv:1011.6646 by
other author
Uniform estimates of nonlinear spectral gaps
By generalizing the path method, we show that nonlinear spectral gaps of a
finite connected graph are uniformly bounded from below by a positive constant
which is independent of the target metric space. We apply our result to an
-ball in the -regular tree, and observe that the asymptotic
behavior of nonlinear spectral gaps of as does not
depend on the target metric space, which is in contrast to the case of a
sequence of expanders. We also apply our result to the -dimensional Hamming
cube and obtain an estimate of its nonlinear spectral gap with respect to
an arbitrary metric space, which is asymptotically sharp as .Comment: to appear in Graphs and Combinatoric
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