Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2402.11222

Treewidth versus clique number. IV. Tree-independence number of graphs excluding an induced star

Clément Dallard·Matjaž Krnc·O-joung Kwon·Martin Milanič·Andrea Munaro·Kenny Štorgel·Sebastian Wiederrecht
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 $(tw,ω)$-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, Milanič, and Štorgel in 2024. Dallard et al. conjectured that $(tw,ω)$-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 $P_4$-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.

Explore related subjects

math.COcs.DMcs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Clément Dallard, Matjaž Krnc, O-joung Kwon, Martin Milanič, Andrea Munaro, Kenny Štorgel, Sebastian Wiederrecht. 2024-02-20. Treewidth versus clique number. IV. Tree-independence number of graphs excluding an induced star. https://arxiv.org/abs/2402.11222

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗