138 research outputs found
On Directed Feedback Vertex Set parameterized by treewidth
We study the Directed Feedback Vertex Set problem parameterized by the
treewidth of the input graph. We prove that unless the Exponential Time
Hypothesis fails, the problem cannot be solved in time on general directed graphs, where is the treewidth of
the underlying undirected graph. This is matched by a dynamic programming
algorithm with running time .
On the other hand, we show that if the input digraph is planar, then the
running time can be improved to .Comment: 20
On the chromatic number of multiple interval graphs and overlap graphs
AbstractLet χ(G) and ω(G) denote the chromatic number and clique number of a graph G. We prove that χ can be bounded by a function of ω for two well-known relatives of interval graphs. Multiple interval graphs (the intersection graphs of sets which can be written as the union of t closed intervals of a line) satisfy χ⩽2t(ω−1) for ω⩾2. Overlap graphs satisfy χ⩽2ωω2(ω−1)
Adding Isolated Vertices Makes some Online Algorithms Optimal
An unexpected difference between online and offline algorithms is observed.
The natural greedy algorithms are shown to be worst case online optimal for
Online Independent Set and Online Vertex Cover on graphs with 'enough' isolated
vertices, Freckle Graphs. For Online Dominating Set, the greedy algorithm is
shown to be worst case online optimal on graphs with at least one isolated
vertex. These algorithms are not online optimal in general. The online
optimality results for these greedy algorithms imply optimality according to
various worst case performance measures, such as the competitive ratio. It is
also shown that, despite this worst case optimality, there are Freckle graphs
where the greedy independent set algorithm is objectively less good than
another algorithm. It is shown that it is NP-hard to determine any of the
following for a given graph: the online independence number, the online vertex
cover number, and the online domination number.Comment: A footnote in the .tex file didn't show up in the last version. This
was fixe
Partitioning the power set of into -free parts
We show that for , in any partition of ,
the set of all subsets of , into parts, some
part must contain a triangle --- three different subsets
such that , , and have distinct representatives.
This is sharp, since by placing two complementary pairs of sets into each
partition class, we have a partition into triangle-free parts. We
also address a more general Ramsey-type problem: for a given graph , find
(estimate) , the smallest number of colors needed for a coloring of
, such that no color class contains a Berge- subhypergraph.
We give an upper bound for for any connected graph which is
asymptotically sharp (for fixed ) when , a cycle, path, or
star with edges. Additional bounds are given for and .Comment: 12 page
Covering t-element Sets by Partitions
Partitions of a set V form a t-cover if each t-element subset is covered by some block of some partitions. The rank of a t-cover is the size of the largest block appearing. What is the minimum rank of a t-cover of an n-element set, consisting of r partitions? The main result says that it is at least n/q, where q is the smallest integer satisfying r ⩽ qt−1 + qt−2 + ⋯ + q + 1
Induced subtrees in graphs of large chromatic number
AbstractOur paper proves special cases of the following conjecture: for any fixed tree T there exists a natural number f = f (T) to that every triangle-free graph of chromaticnumber f(T) contains T as an induced subgraph. The main result concerns the case when T has radius two
Clique-cutsets beyond chordal graphs
Truemper configurations (thetas, pyramids, prisms, and wheels) have played an important role in the study of complex hereditary graph classes (eg, the class of perfect graphs and the class of even‐hole‐free graphs), appearing both as excluded configurations, and as configurations around which graphs can be decomposed. In this paper, we study the structure of graphs that contain (as induced subgraphs) no Truemper configurations other than (possibly) universal wheels and twin wheels. We also study several subclasses of this class. We use our structural results to analyze the complexity of the recognition, maximum weight clique, maximum weight stable set, and optimal vertex coloring problems for these classes. Furthermore, we obtain polynomial x-bounding functions for these classes
Approximately coloring graphs without long induced paths
It is an open problem whether the 3-coloring problem can be solved in
polynomial time in the class of graphs that do not contain an induced path on
vertices, for fixed . We propose an algorithm that, given a 3-colorable
graph without an induced path on vertices, computes a coloring with
many colors. If the input graph is
triangle-free, we only need many
colors. The running time of our algorithm is if the input
graph has vertices and edges
Coloring translates and homothets of a convex body
We obtain improved upper bounds and new lower bounds on the chromatic number
as a linear function of the clique number, for the intersection graphs (and
their complements) of finite families of translates and homothets of a convex
body in \RR^n.Comment: 11 pages, 2 figure
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