105,564 research outputs found
Covering Small Independent Sets and Separators with Applications to Parameterized Algorithms
We present two new combinatorial tools for the design of parameterized
algorithms. The first is a simple linear time randomized algorithm that given
as input a -degenerate graph and an integer , outputs an independent
set , such that for every independent set in of size at most ,
the probability that is a subset of is at least .The second is a new (deterministic) polynomial
time graph sparsification procedure that given a graph , a set of terminal pairs and an
integer , returns an induced subgraph of that maintains all
the inclusion minimal multicuts of of size at most , and does not
contain any -vertex connected set of size . In
particular, excludes a clique of size as a
topological minor. Put together, our new tools yield new randomized fixed
parameter tractable (FPT) algorithms for Stable - Separator, Stable Odd
Cycle Transversal and Stable Multicut on general graphs, and for Stable
Directed Feedback Vertex Set on -degenerate graphs, resolving two problems
left open by Marx et al. [ACM Transactions on Algorithms, 2013]. All of our
algorithms can be derandomized at the cost of a small overhead in the running
time.Comment: 35 page
An algebraic formulation of the graph reconstruction conjecture
The graph reconstruction conjecture asserts that every finite simple graph on
at least three vertices can be reconstructed up to isomorphism from its deck -
the collection of its vertex-deleted subgraphs. Kocay's Lemma is an important
tool in graph reconstruction. Roughly speaking, given the deck of a graph
and any finite sequence of graphs, it gives a linear constraint that every
reconstruction of must satisfy.
Let be the number of distinct (mutually non-isomorphic) graphs on
vertices, and let be the number of distinct decks that can be
constructed from these graphs. Then the difference measures
how many graphs cannot be reconstructed from their decks. In particular, the
graph reconstruction conjecture is true for -vertex graphs if and only if
.
We give a framework based on Kocay's lemma to study this discrepancy. We
prove that if is a matrix of covering numbers of graphs by sequences of
graphs, then . In particular, all
-vertex graphs are reconstructible if one such matrix has rank . To
complement this result, we prove that it is possible to choose a family of
sequences of graphs such that the corresponding matrix of covering numbers
satisfies .Comment: 12 pages, 2 figure
Disproving the normal graph conjecture
A graph is called normal if there exist two coverings, and
of its vertex set such that every member of induces a
clique in , every member of induces an independent set in
and for every and . It has been conjectured by De Simone and K\"orner in 1999 that a
graph is normal if does not contain , and
as an induced subgraph. We disprove this conjecture
Symmetry adapted Assur decompositions
Assur graphs are a tool originally developed by mechanical engineers to
decompose mechanisms for simpler analysis and synthesis. Recent work has
connected these graphs to strongly directed graphs, and decompositions of the
pinned rigidity matrix. Many mechanisms have initial configurations which are
symmetric, and other recent work has exploited the orbit matrix as a symmetry
adapted form of the rigidity matrix. This paper explores how the decomposition
and analysis of symmetric frameworks and their symmetric motions can be
supported by the new symmetry adapted tools.Comment: 40 pages, 22 figure
The Relativized Second Eigenvalue Conjecture of Alon
We prove a relativization of the Alon Second Eigenvalue Conjecture for all
-regular base graphs, , with : for any , we show that
a random covering map of degree to has a new eigenvalue greater than
in absolute value with probability .
Furthermore, if is a Ramanujan graph, we show that this probability is
proportional to , where
is an integer depending on , which can be computed by a finite algorithm for
any fixed . For any -regular graph, , is
greater than .
Our proof introduces a number of ideas that simplify and strengthen the
methods of Friedman's proof of the original conjecture of Alon. The most
significant new idea is that of a ``certified trace,'' which is not only
greatly simplifies our trace methods, but is the reason we can obtain the
estimate above. This estimate represents an
improvement over Friedman's results of the original Alon conjecture for random
-regular graphs, for certain values of
Covering Pairs in Directed Acyclic Graphs
The Minimum Path Cover problem on directed acyclic graphs (DAGs) is a
classical problem that provides a clear and simple mathematical formulation for
several applications in different areas and that has an efficient algorithmic
solution. In this paper, we study the computational complexity of two
constrained variants of Minimum Path Cover motivated by the recent introduction
of next-generation sequencing technologies in bioinformatics. The first problem
(MinPCRP), given a DAG and a set of pairs of vertices, asks for a minimum
cardinality set of paths "covering" all the vertices such that both vertices of
each pair belong to the same path. For this problem, we show that, while it is
NP-hard to compute if there exists a solution consisting of at most three
paths, it is possible to decide in polynomial time whether a solution
consisting of at most two paths exists. The second problem (MaxRPSP), given a
DAG and a set of pairs of vertices, asks for a path containing the maximum
number of the given pairs of vertices. We show its NP-hardness and also its
W[1]-hardness when parametrized by the number of covered pairs. On the positive
side, we give a fixed-parameter algorithm when the parameter is the maximum
overlapping degree, a natural parameter in the bioinformatics applications of
the problem
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