68,496 research outputs found
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
A simple linear time algorithm for the locally connected spanning tree problem on maximal planar chordal graphs
A locally connected spanning tree (LCST) T of a graph G is a spanning tree of G such that, for each node, its neighborhood in T induces a connected sub- graph in G. The problem of determining whether a graph contains an LCST or not has been proved to be NP-complete, even if the graph is planar or chordal. The main result of this paper is a simple linear time algorithm that, given a maximal planar chordal graph, determines in linear time whether it contains an LCST or not, and produces one if it exists. We give an anal- ogous result for the case when the input graph is a maximal outerplanar graph
Minimal Obstructions for Partial Representations of Interval Graphs
Interval graphs are intersection graphs of closed intervals. A generalization
of recognition called partial representation extension was introduced recently.
The input gives an interval graph with a partial representation specifying some
pre-drawn intervals. We ask whether the remaining intervals can be added to
create an extending representation. Two linear-time algorithms are known for
solving this problem.
In this paper, we characterize the minimal obstructions which make partial
representations non-extendible. This generalizes Lekkerkerker and Boland's
characterization of the minimal forbidden induced subgraphs of interval graphs.
Each minimal obstruction consists of a forbidden induced subgraph together with
at most four pre-drawn intervals. A Helly-type result follows: A partial
representation is extendible if and only if every quadruple of pre-drawn
intervals is extendible by itself. Our characterization leads to a linear-time
certifying algorithm for partial representation extension
Restricted Space Algorithms for Isomorphism on Bounded Treewidth Graphs
The Graph Isomorphism problem restricted to graphs of bounded treewidth or
bounded tree distance width are known to be solvable in polynomial time
[Bod90],[YBFT99]. We give restricted space algorithms for these problems
proving the following results: - Isomorphism for bounded tree distance width
graphs is in L and thus complete for the class. We also show that for this kind
of graphs a canon can be computed within logspace. - For bounded treewidth
graphs, when both input graphs are given together with a tree decomposition,
the problem of whether there is an isomorphism which respects the
decompositions (i.e. considering only isomorphisms mapping bags in one
decomposition blockwise onto bags in the other decomposition) is in L. - For
bounded treewidth graphs, when one of the input graphs is given with a tree
decomposition the isomorphism problem is in LogCFL. - As a corollary the
isomorphism problem for bounded treewidth graphs is in LogCFL. This improves
the known TC1 upper bound for the problem given by Grohe and Verbitsky
[GroVer06].Comment: STACS conference 2010, 12 page
Unique Perfect Phylogeny Characterizations via Uniquely Representable Chordal Graphs
The perfect phylogeny problem is a classic problem in computational biology,
where we seek an unrooted phylogeny that is compatible with a set of
qualitative characters. Such a tree exists precisely when an intersection graph
associated with the character set, called the partition intersection graph, can
be triangulated using a restricted set of fill edges. Semple and Steel used the
partition intersection graph to characterize when a character set has a unique
perfect phylogeny. Bordewich, Huber, and Semple showed how to use the partition
intersection graph to find a maximum compatible set of characters. In this
paper, we build on these results, characterizing when a unique perfect
phylogeny exists for a subset of partial characters. Our characterization is
stated in terms of minimal triangulations of the partition intersection graph
that are uniquely representable, also known as ur-chordal graphs. Our
characterization is motivated by the structure of ur-chordal graphs, and the
fact that the block structure of minimal triangulations is mirrored in the
graph that has been triangulated
A Polynomial Delay Algorithm for Enumerating Minimal Dominating Sets in Chordal Graphs
An output-polynomial algorithm for the listing of minimal dominating sets in
graphs is a challenging open problem and is known to be equivalent to the
well-known Transversal problem which asks for an output-polynomial algorithm
for listing the set of minimal hitting sets in hypergraphs. We give a
polynomial delay algorithm to list the set of minimal dominating sets in
chordal graphs, an important and well-studied graph class where such an
algorithm was open for a while.Comment: 13 pages, 1 figure, submitte
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