184 research outputs found
On the threshold-width of graphs
The GG-width of a class of graphs GG is defined as follows. A graph G has
GG-width k if there are k independent sets N1,...,Nk in G such that G can be
embedded into a graph H in GG such that for every edge e in H which is not an
edge in G, there exists an i such that both endpoints of e are in Ni. For the
class TH of threshold graphs we show that TH-width is NP-complete and we
present fixed-parameter algorithms. We also show that for each k, graphs of
TH-width at most k are characterized by a finite collection of forbidden
induced subgraphs
Fast algorithms for the Tron game on trees
Abstract TR.ON is the following game which can be played on any graph: Two players choose alternately a node of the graph subject to the requirement that each player must choose a node which is adjacent to his previously chosen node and such that every node is chosen only once. In this paper O(n) and O(ny'n) algorithms are given for deciding whether there is a winning strategy for the first player when TR.oN is played on a given tree, for the variants with and without specified starting nodes, respectively. The problem is shown to be both NP-hard and co-NP-hard for connected undirected graphs in general
On the stable degree of graphs
We define the stable degree s(G) of a graph G by s(G)∈=∈ min max d (v), where the minimum is taken over all maximal independent sets U of G. For this new parameter we prove the following. Deciding whether a graph has stable degree at most k is NP-complete for every fixed k∈≥∈3; and the stable degree is hard to approximate. For asteroidal triple-free graphs and graphs of bounded asteroidal number the stable degree can be computed in polynomial time. For graphs in these classes the treewidth is bounded from below and above in terms of the stable degree
Efficient FPT algorithms for (strict) compatibility of unrooted phylogenetic trees
In phylogenetics, a central problem is to infer the evolutionary
relationships between a set of species ; these relationships are often
depicted via a phylogenetic tree -- a tree having its leaves univocally labeled
by elements of and without degree-2 nodes -- called the "species tree". One
common approach for reconstructing a species tree consists in first
constructing several phylogenetic trees from primary data (e.g. DNA sequences
originating from some species in ), and then constructing a single
phylogenetic tree maximizing the "concordance" with the input trees. The
so-obtained tree is our estimation of the species tree and, when the input
trees are defined on overlapping -- but not identical -- sets of labels, is
called "supertree". In this paper, we focus on two problems that are central
when combining phylogenetic trees into a supertree: the compatibility and the
strict compatibility problems for unrooted phylogenetic trees. These problems
are strongly related, respectively, to the notions of "containing as a minor"
and "containing as a topological minor" in the graph community. Both problems
are known to be fixed-parameter tractable in the number of input trees , by
using their expressibility in Monadic Second Order Logic and a reduction to
graphs of bounded treewidth. Motivated by the fact that the dependency on
of these algorithms is prohibitively large, we give the first explicit dynamic
programming algorithms for solving these problems, both running in time
, where is the total size of the input.Comment: 18 pages, 1 figur
Treewidth and minimum fill-in on d-trapezoid graphs
We show that the minimum fill-in and the minimum interval graph completion of a d-trapezoid graph can be computed in time On d . We also show that the treewidth and the pathwidth of a d-trapezoid graph can be computed by an On twG d,1 time algorithm. For both algorithms, d is supposed to be a fixed positive integer and it is required that a suitable intersection model of the given d-trapezoid graph is part of the input. As a consequence, the minimum fill-in and the minimum interval graph completion as well as the treewidth and the pathwidth of a given trapezoid graph (or permutation graph) can be computed in time On 2 , even if no intersection model is part of the input
Line-distortion, Bandwidth and Path-length of a graph
We investigate the minimum line-distortion and the minimum bandwidth problems
on unweighted graphs and their relations with the minimum length of a
Robertson-Seymour's path-decomposition. The length of a path-decomposition of a
graph is the largest diameter of a bag in the decomposition. The path-length of
a graph is the minimum length over all its path-decompositions. In particular,
we show:
- if a graph can be embedded into the line with distortion , then
admits a Robertson-Seymour's path-decomposition with bags of diameter at most
in ;
- for every class of graphs with path-length bounded by a constant, there
exist an efficient constant-factor approximation algorithm for the minimum
line-distortion problem and an efficient constant-factor approximation
algorithm for the minimum bandwidth problem;
- there is an efficient 2-approximation algorithm for computing the
path-length of an arbitrary graph;
- AT-free graphs and some intersection families of graphs have path-length at
most 2;
- for AT-free graphs, there exist a linear time 8-approximation algorithm for
the minimum line-distortion problem and a linear time 4-approximation algorithm
for the minimum bandwidth problem
An FPT 2-Approximation for Tree-Cut Decomposition
The tree-cut width of a graph is a graph parameter defined by Wollan [J.
Comb. Theory, Ser. B, 110:47-66, 2015] with the help of tree-cut
decompositions. In certain cases, tree-cut width appears to be more adequate
than treewidth as an invariant that, when bounded, can accelerate the
resolution of intractable problems. While designing algorithms for problems
with bounded tree-cut width, it is important to have a parametrically tractable
way to compute the exact value of this parameter or, at least, some constant
approximation of it. In this paper we give a parameterized 2-approximation
algorithm for the computation of tree-cut width; for an input -vertex graph
and an integer , our algorithm either confirms that the tree-cut width
of is more than or returns a tree-cut decomposition of certifying
that its tree-cut width is at most , in time .
Prior to this work, no constructive parameterized algorithms, even approximated
ones, existed for computing the tree-cut width of a graph. As a consequence of
the Graph Minors series by Robertson and Seymour, only the existence of a
decision algorithm was known.Comment: 17 pages, 3 figure
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