3 research outputs found
Treewidth versus clique number. II. Tree-independence number
In 2020, we initiated a systematic study of graph classes in which the
treewidth can only be large due to the presence of a large clique, which we
call -bounded. While -bounded graph
classes are known to enjoy some good algorithmic properties related to clique
and coloring problems, it is an interesting open problem whether
-boundedness also has useful algorithmic implications for
problems related to independent sets.
We provide a partial answer to this question by means of a new min-max graph
invariant related to tree decompositions. We define the independence number of
a tree decomposition of a graph as the maximum independence
number over all subgraphs of induced by some bag of . The
tree-independence number of a graph is then defined as the minimum
independence number over all tree decompositions of . Generalizing a result
on chordal graphs due to Cameron and Hell from 2006, we show that if a graph is
given together with a tree decomposition with bounded independence number, then
the Maximum Weight Independent Packing problem can be solved in polynomial
time.
Applications of our general algorithmic result to specific graph classes will
be given in the third paper of the series [Dallard, Milani\v{c}, and
\v{S}torgel, Treewidth versus clique number. III. Tree-independence number of
graphs with a forbidden structure].Comment: 33 pages; abstract has been shortened due to arXiv requirements. A
previous version of this arXiv post has been reorganized into two parts; this
is the first of the two parts (the second one is arXiv:2206.15092
Three-in-a-Tree in Near Linear Time
The three-in-a-tree problem is to determine if a simple undirected graph
contains an induced subgraph which is a tree connecting three given vertices.
Based on a beautiful characterization that is proved in more than twenty pages,
Chudnovsky and Seymour [Combinatorica 2010] gave the previously only known
polynomial-time algorithm, running in time, to solve the
three-in-a-tree problem on an -vertex -edge graph. Their three-in-a-tree
algorithm has become a critical subroutine in several state-of-the-art graph
recognition and detection algorithms.
In this paper we solve the three-in-a-tree problem in time,
leading to improved algorithms for recognizing perfect graphs and detecting
thetas, pyramids, beetles, and odd and even holes. Our result is based on a new
and more constructive characterization than that of Chudnovsky and Seymour. Our
new characterization is stronger than the original, and our proof implies a new
simpler proof for the original characterization. The improved characterization
gains the first factor in speed. The remaining improvement is based on
dynamic graph algorithms.Comment: 46 pages, 12 figures, accepted to STOC 202