Graph properties link forbidden structures to tree decompositions
Induced packing treewidth II. Excluding a clique or a biclique
Discrete Mathematics
Summary
This paper studies a way to measure the complexity of graphs by counting specific small patterns inside them, extending previous methods. The authors show that if certain large complete bipartite graphs are not allowed, graphs with limited complexity in one sense must also have limited complexity in another. They also prove that classes of graphs with restricted small patterns have well-behaved coloring properties. Finally, they relate this new measurement to another graph parameter and solve questions that connect these ideas.
What this means in practice
- •For graph algorithm designers: Use results to improve algorithms exploiting bounded complexity in graphs without certain subgraphs, aiding faster graph-based computations.
- •For network analysts: Identify network structures with bounded induced packing treewidth to simplify analysis and visualization of complex networks.
A theory result. No direct application yet.
Authors
Amir Nikabadi, Paweł Rzążewski
Abstract
The notion of induced packing treewidth aims to unify classes defined by forbidden induced subgraphs or induced minors with classes defined by the existence of certain structured tree decompositions. For a graph $H$, \emph{induced $H$-packing treewidth}, denoted by $\treepi_{H}$, is a tree-decomposition-based graph parameter that, for each bag, measures the maximum number of pairwise anticomplete induced copies of $H$ intersecting that bag. This notion generalizes some previously studied parameters: when $H=P_1$, it is equivalent to tree-independence number, and when $H=P_2$, it is equivalent to induced matching treewidth. We prove the following: \begin{itemize}[itemsep=2mm,leftmargin=6mm] \item For all $a,t\in \mathbb{N}$, $K_{a,a}$-free graphs of bounded induced $P_t$-packing treewidth have bounded tree-independence number. This extends the previous result of Abrishami et al. [SIAM J. Discrete Math., 2025] for $t=2$, and a result of Hajebi and Spirkl who showed that $(P_t,K_{a,a})$-free graphs have bounded tree-independence number. \item If $H$ is any fixed path or a star, then the class of graphs of bounded induced $H$-packing treewidth is $χ$-bounded. Again, this extends the previous result of Abrishami et al. [SIAM J. Discrete Math., 2025] for $H=P_2$. \item Finally, we study the relationship between induced packing treewidth and \emph{sim-width}, a width parameter based on branch decompositions. We show that, although \emph{sim-width} and induced $P_3$-packing treewidth are incomparable, graphs of bounded sim-width that exclude all \emph{$H$-obstructions}---certain graphs that force large induced $H$-packing treewidth---have bounded induced $H$-packing treewidth. This simultaneously generalizes and resolves questions posed by Abrishami et al. [SIAM J. Discrete Math., 2025] and Brettell et al. [European J. Comb., 2025]. \end{itemize}