22 research outputs found
Tree Deletion Set has a Polynomial Kernel (but no OPT^O(1) approximation)
In the Tree Deletion Set problem the input is a graph G together with an
integer k. The objective is to determine whether there exists a set S of at
most k vertices such that G-S is a tree. The problem is NP-complete and even
NP-hard to approximate within any factor of OPT^c for any constant c. In this
paper we give a O(k^4) size kernel for the Tree Deletion Set problem. To the
best of our knowledge our result is the first counterexample to the
"conventional wisdom" that kernelization algorithms automatically provide
approximation algorithms with approximation ratio close to the size of the
kernel. An appealing feature of our kernelization algorithm is a new algebraic
reduction rule that we use to handle the instances on which Tree Deletion Set
is hard to approximate
Forbidding Kuratowski Graphs as Immersions
The immersion relation is a partial ordering relation on graphs that is
weaker than the topological minor relation in the sense that if a graph
contains a graph as a topological minor, then it also contains it as an
immersion but not vice versa. Kuratowski graphs, namely and ,
give a precise characterization of planar graphs when excluded as topological
minors. In this note we give a structural characterization of the graphs that
exclude Kuratowski graphs as immersions. We prove that they can be constructed
by applying consecutive -edge-sums, for , starting from graphs that
are planar sub-cubic or of branch-width at most 10
Cutwidth: obstructions and algorithmic aspects
Cutwidth is one of the classic layout parameters for graphs. It measures how
well one can order the vertices of a graph in a linear manner, so that the
maximum number of edges between any prefix and its complement suffix is
minimized. As graphs of cutwidth at most are closed under taking
immersions, the results of Robertson and Seymour imply that there is a finite
list of minimal immersion obstructions for admitting a cut layout of width at
most . We prove that every minimal immersion obstruction for cutwidth at
most has size at most .
As an interesting algorithmic byproduct, we design a new fixed-parameter
algorithm for computing the cutwidth of a graph that runs in time , where is the optimum width and is the number of vertices.
While being slower by a -factor in the exponent than the fastest known
algorithm, given by Thilikos, Bodlaender, and Serna in [Cutwidth I: A linear
time fixed parameter algorithm, J. Algorithms, 56(1):1--24, 2005] and [Cutwidth
II: Algorithms for partial -trees of bounded degree, J. Algorithms,
56(1):25--49, 2005], our algorithm has the advantage of being simpler and
self-contained; arguably, it explains better the combinatorics of optimum-width
layouts
Forbidding Kuratowski graphs as immersions
The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph G contains a graph H as a topological minor, then it also contains it as an immersion but not vice versa. Kuratowski graphs, namely K 5 and K 3,3 , give a precise characterization of planar graphs when excluded as topological minors. In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions. We prove that they can be constructed by applying consecutive i-edge-sums, for i ≤ 3, starting from graphs that are planar sub-cubic or of branchwidth at most 10