53,293 research outputs found
On Tree-Partition-Width
A \emph{tree-partition} of a graph is a proper partition of its vertex
set into `bags', such that identifying the vertices in each bag produces a
forest. The \emph{tree-partition-width} of is the minimum number of
vertices in a bag in a tree-partition of . An anonymous referee of the paper
by Ding and Oporowski [\emph{J. Graph Theory}, 1995] proved that every graph
with tree-width and maximum degree has
tree-partition-width at most . We prove that this bound is within a
constant factor of optimal. In particular, for all and for all
sufficiently large , we construct a graph with tree-width , maximum
degree , and tree-partition-width at least (\eighth-\epsilon)k\Delta.
Moreover, we slightly improve the upper bound to
without the restriction that
On the Parameterized Complexity of Computing Tree-Partitions
We study the parameterized complexity of computing the tree-partition-width,
a graph parameter equivalent to treewidth on graphs of bounded maximum degree.
On one hand, we can obtain approximations of the tree-partition-width
efficiently: we show that there is an algorithm that, given an -vertex graph
and an integer , constructs a tree-partition of width for
or reports that has tree-partition width more than , in time
. We can improve on the approximation factor or the dependence on
by sacrificing the dependence on .
On the other hand, we show the problem of computing tree-partition-width
exactly is XALP-complete, which implies that it is -hard for all . We
deduce XALP-completeness of the problem of computing the domino treewidth.
Finally, we adapt some known results on the parameter tree-partition-width and
the topological minor relation, and use them to compare tree-partition-width to
tree-cut width
On the parameterized complexity of computing tree-partitions
We study the parameterized complexity of computing the tree-partition-width,
a graph parameter equivalent to treewidth on graphs of bounded maximum degree.
On one hand, we can obtain approximations of the tree-partition-width
efficiently: we show that there is an algorithm that, given an -vertex graph
and an integer , constructs a tree-partition of width for
or reports that has tree-partition width more than , in time
. We can improve on the approximation factor or the dependence on
by sacrificing the dependence on .
On the other hand, we show the problem of computing tree-partition-width
exactly is XALP-complete, which implies that it is -hard for all . We
deduce XALP-completeness of the problem of computing the domino treewidth.
Finally, we adapt some known results on the parameter tree-partition-width and
the topological minor relation, and use them to compare tree-partition-width to
tree-cut width
On the Parameterized Complexity of Computing Tree-Partitions
We study the parameterized complexity of computing the tree-partition-width, a graph parameter equivalent to treewidth on graphs of bounded maximum degree. On one hand, we can obtain approximations of the tree-partition-width efficiently: we show that there is an algorithm that, given an n-vertex graph G and an integer k, constructs a tree-partition of width O(k7) for G or reports that G has tree-partition width more than k, in time kO(1)n2. We can improve slightly on the approximation factor by sacrificing the dependence on k, or on n. On the other hand, we show the problem of computing tree-partition-width exactly is XALP-complete, which implies that it is W[t]-hard for all t. We deduce XALP-completeness of the problem of computing the domino treewidth. Finally, we adapt some known results on the parameter tree-partition-width and the topological minor relation, and use them to compare tree-partition-width to tree-cut width
A new width parameter of graphs based on edge cuts: -edge-crossing width
We introduce graph width parameters, called -edge-crossing width and
edge-crossing width. These are defined in terms of the number of edges crossing
a bag of a tree-cut decomposition. They are motivated by edge-cut width,
recently introduced by Brand et al. (WG 2022). We show that edge-crossing width
is equivalent to the known parameter tree-partition-width. On the other hand,
-edge-crossing width is a new parameter; tree-cut width and
-edge-crossing width are incomparable, and they both lie between
tree-partition-width and edge-cut width.
We provide an algorithm that, for a given -vertex graph and integers
and , in time either outputs
a tree-cut decomposition certifying that the -edge-crossing width of
is at most or confirms that the -edge-crossing width
of is more than . As applications, for every fixed , we obtain
FPT algorithms for the List Coloring and Precoloring Extension problems
parameterized by -edge-crossing width. They were known to be W[1]-hard
parameterized by tree-partition-width, and FPT parameterized by edge-cut width,
and we close the complexity gap between these two parameters.Comment: 26 pages, 1 figure, accepted to WG202
Grad and classes with bounded expansion I. decompositions
We introduce classes of graphs with bounded expansion as a generalization of
both proper minor closed classes and degree bounded classes. Such classes are
based on a new invariant, the greatest reduced average density (grad) of G with
rank r, grad r(G). For these classes we prove the existence of several
partition results such as the existence of low tree-width and low tree-depth
colorings. This generalizes and simplifies several earlier results (obtained
for minor closed classes)
Individual tree measurements by means of digital aerial photogrammetry
Korpela, I. 2004. Individual tree measurements by means of digital aerial photogrammetry. Silva Fennica Monographs 3. 93 p. This study explores the plausibility of the use of multi-scale, CIR aerial photographs to conduct forest inventory at the individual tree level. Multiple digitised aerial photographs are used for manual and semi-automatic 3D positioning of tree tops, for species classification, and for measurements on tree height and crown width. A new tree top positioning algorithm is presented and tested. It incorporates template matching in a 3D search space. Also, a new method is presented for tree species classification. In it, a partition of the image space according to the continuously varying image-object-sun geometry of aerial views is performed. Discernibility of trees in aerial images is studied. The measurement accuracy and overall measurability of crown width by using manual image measurements is investigated. A simulation study is used to examine the combined effects of discernibility and photogrammetric measurement errors on stand variables. The study material contained large-scale colour and CIR image material and 7708 trees from 24 fully mappe
The Parameterised Complexity of Integer Multicommodity Flow
The Integer Multicommodity Flow problem has been studied extensively in the
literature. However, from a parameterised perspective, mostly special cases,
such as the Disjoint Paths problem, have been considered. Therefore, we
investigate the parameterised complexity of the general Integer Multicommodity
Flow problem. We show that the decision version of this problem on directed
graphs for a constant number of commodities, when the capacities are given in
unary, is XNLP-complete with pathwidth as parameter and XALP-complete with
treewidth as parameter. When the capacities are given in binary, the problem is
NP-complete even for graphs of pathwidth at most 13. We give related results
for undirected graphs. These results imply that the problem is unlikely to be
fixed-parameter tractable by these parameters.
In contrast, we show that the problem does become fixed-parameter tractable
when weighted tree partition width (a variant of tree partition width for edge
weighted graphs) is used as parameter
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