6 research outputs found
New Width Parameters for Independent Set: One-sided-mim-width and Neighbor-depth
We study the tractability of the maximum independent set problem from the
viewpoint of graph width parameters, with the goal of defining a width
parameter that is as general as possible and allows to solve independent set in
polynomial-time on graphs where the parameter is bounded. We introduce two new
graph width parameters: one-sided maximum induced matching-width (o-mim-width)
and neighbor-depth. O-mim-width is a graph parameter that is more general than
the known parameters mim-width and tree-independence number, and we show that
independent set and feedback vertex set can be solved in polynomial-time given
a decomposition with bounded o-mim-width. O-mim-width is the first width
parameter that gives a common generalization of chordal graphs and graphs of
bounded clique-width in terms of tractability of these problems.
The parameter o-mim-width, as well as the related parameters mim-width and
sim-width, have the limitation that no algorithms are known to compute
bounded-width decompositions in polynomial-time. To partially resolve this
limitation, we introduce the parameter neighbor-depth. We show that given a
graph of neighbor-depth , independent set can be solved in time
even without knowing a corresponding decomposition. We also show that
neighbor-depth is bounded by a polylogarithmic function on the number of
vertices on large classes of graphs, including graphs of bounded o-mim-width,
and more generally graphs of bounded sim-width, giving a quasipolynomial-time
algorithm for independent set on these graph classes. This resolves an open
problem asked by Kang, Kwon, Str{\o}mme, and Telle [TCS 2017].Comment: 26 pages, 1 figure. This is the full version of an extended abstract
that will appear in WG202
Max Weight Independent Set in sparse graphs with no long claws
We revisit the recent polynomial-time algorithm for the MAX WEIGHT
INDEPENDENT SET (MWIS) problem in bounded-degree graphs that do not contain a
fixed graph whose every component is a subdivided claw as an induced subgraph
[Abrishami, Dibek, Chudnovsky, Rz\k{a}\.zewski, SODA 2022].
First, we show that with an arguably simpler approach we can obtain a faster
algorithm with running time , where is the
number of vertices of the instance and is the maximum degree. Then we
combine our technique with known results concerning tree decompositions and
provide a polynomial-time algorithm for MWIS in graphs excluding a fixed graph
whose every component is a subdivided claw as an induced subgraph, and a fixed
biclique as a subgraph
Odd-Minors I: Excluding small parity breaks
Given a graph class~, the -blind-treewidth of a
graph~ is the smallest integer~ such that~ has a tree-decomposition
where every bag whose torso does not belong to~ has size at
most~. In this paper we focus on the class~ of bipartite graphs
and the class~ of planar graphs together with the odd-minor
relation. For each of the two parameters, -blind-treewidth and
-blind-treewidth, we prove an analogue of the
celebrated Grid Theorem under the odd-minor relation. As a consequence we
obtain FPT-approximation algorithms for both parameters. We then provide
FPT-algorithms for \textsc{Maximum Independent Set} on graphs of bounded
-blind-treewidth and \textsc{Maximum Cut} on graphs of bounded
-blind-treewidth
Treewidth versus clique number. IV. Tree-independence number of graphs excluding an induced star
Many recent works address the question of characterizing induced obstructions
to bounded treewidth. In 2022, Lozin and Razgon completely answered this
question for graph classes defined by finitely many forbidden induced
subgraphs. Their result also implies a characterization of graph classes
defined by finitely many forbidden induced subgraphs that are
-bounded, that is, treewidth can only be large due to the presence
of a large clique. This condition is known to be satisfied for any graph class
with bounded tree-independence number, a graph parameter introduced
independently by Yolov in 2018 and by Dallard, Milani\v{c}, and \v{S}torgel in
2024. Dallard et al. conjectured that -boundedness is actually
equivalent to bounded tree-independence number. We address this conjecture in
the context of graph classes defined by finitely many forbidden induced
subgraphs and prove it for the case of graph classes excluding an induced star.
We also prove it for subclasses of the class of line graphs, determine the
exact values of the tree-independence numbers of line graphs of complete graphs
and line graphs of complete bipartite graphs, and characterize the
tree-independence number of -free graphs, which implies a linear-time
algorithm for its computation. Applying the algorithmic framework provided in a
previous paper of the series leads to polynomial-time algorithms for the
Maximum Weight Independent Set problem in an infinite family of graph classes.Comment: 26 page
Comparing Width Parameters on Graph Classes
We study how the relationship between non-equivalent width parameters changes
once we restrict to some special graph class. As width parameters, we consider
treewidth, clique-width, twin-width, mim-width, sim-width and tree-independence
number, whereas as graph classes we consider -subgraph-free graphs,
line graphs and their common superclass, for , of -free
graphs.
We first provide a complete comparison when restricted to
-subgraph-free graphs, showing in particular that treewidth,
clique-width, mim-width, sim-width and tree-independence number are all
equivalent. This extends a result of Gurski and Wanke (2000) stating that
treewidth and clique-width are equivalent for the class of
-subgraph-free graphs.
Next, we provide a complete comparison when restricted to line graphs,
showing in particular that, on any class of line graphs, clique-width,
mim-width, sim-width and tree-independence number are all equivalent, and
bounded if and only if the class of root graphs has bounded treewidth. This
extends a result of Gurski and Wanke (2007) stating that a class of graphs
has bounded treewidth if and only if the class of line graphs of
graphs in has bounded clique-width.
We then provide an almost-complete comparison for -free graphs,
leaving one missing case. Our main result is that -free graphs of
bounded mim-width have bounded tree-independence number. This result has
structural and algorithmic consequences. In particular, it proves a special
case of a conjecture of Dallard, Milani\v{c} and \v{S}torgel.
Finally, we consider the question of whether boundedness of a certain width
parameter is preserved under graph powers. We show that the question has a
positive answer for sim-width precisely in the case of odd powers.Comment: 31 pages, 4 figures, abstract shortened due to arXiv requirement
Treewidth versus clique number. III. Tree-independence number of graphs with a forbidden structure
We continue the study of -bounded graph classes, that
is, hereditary graph classes in which the treewidth can only be large due to
the presence of a large clique, with the goal of understanding the extent to
which this property has useful algorithmic implications for the Independent Set
and related problems. In the previous paper of the series [Dallard,
Milani\v{c}, and \v{S}torgel, Treewidth versus clique number. II.
Tree-independence number], we introduced the tree-independence number, a
min-max graph invariant related to tree decompositions. Bounded
tree-independence number implies both -boundedness and
the existence of a polynomial-time algorithm for the Maximum Weight Independent
Set problem, provided that the input graph is given together with a tree
decomposition with bounded independence number.
In this paper, we consider six graph containment relations and for each of
them characterize the graphs for which any graph excluding with respect
to the relation admits a tree decomposition with bounded independence number.
The induced minor relation is of particular interest: we show that excluding
either a minus an edge or the -wheel implies the existence of a tree
decomposition in which every bag is a clique plus at most vertices, while
excluding a complete bipartite graph implies the existence of a tree
decomposition with independence number at most . Our constructive
proofs are obtained using a variety of tools, including -refined tree
decompositions, SPQR trees, and potential maximal cliques. They imply
polynomial-time algorithms for the Independent Set and related problems in an
infinite family of graph classes; in particular, the results apply to the class
of -perfectly orientable graphs, answering a question of Beisegel,
Chudnovsky, Gurvich, Milani\v{c}, and Servatius from 2019.Comment: 46 pages; abstract has been shortened due to arXiv requirements. A
previous arXiv post (arXiv:2111.04543) has been reorganized into two parts;
this is the second of the two part