7 research outputs found
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