In 2020, Dallard, Milanič, and Štorgel initiated a systematic study of graph classes in which the treewidth can only be large due the presence of a large clique, which they call (tw, ω)bounded. The family of (tw, ω)-bounded graph classes provides a unifying framework for a variety of very different families of graph classes, including graph classes of bounded treewidth, graph classes of bounded independence number, intersection graphs of connected subgraphs of graphs with bounded treewidth, and graphs in which all minimal separators are of bounded size. While Chaplick and Zeman showed in 2017 that (tw, ω)-bounded graph classes enjoy some good algorithmic properties related to clique and coloring problems, an interesting open problem is whether (tw, ω)-boundedness has useful algorithmic implications for problems related to independent sets. We provide a partial answer to this question by identifying a sufficient condition for (tw, ω)-bounded graph classes to admit a polynomial-time algorithm for the Maximum Weight Independent H-Packing problem, for any fixed finite set H of connected graphs. This family of problems generalizes several other problems studied in the literature, including the Maximum Weight Independent Set and Maximum Weight Induced Matching problems.Our approach is based on a new min-max graph parameter related to tree decompositions. We define the independence number of a tree decomposition T of a graph as the maximum independence number over all subgraphs of G induced by some bag of T . The tree-independence number of a graph G is then defined as the minimum independence number over all tree decompositions of G. Boundedness of the tree-independence number is a refinement of (tw, ω)-boundedness that is still general enough to hold for all the aforementioned families of graph classes. We show that if a graph is given together with a tree decomposition with bounded independence number, then for any fixed finite set H of connected graphs, the Maximum Weight Independent H-Packing problem can be solved in polynomial time. Motivated by this result, we consider six graph containment relations-the subgraph, topological minor, and minor relations, as well as their induced variants-and for each of them characterize the graphs H for which any graph excluding H with respect to the relation admits a tree decomposition with bounded independence number. These results build on and refine the analogous characterizations due to Dallard, Milanič, and Štorgel for (tw, ω)-boundedness, and shows that in all these cases, (tw, ω)boundedness is actually equivalent to bounded tree-independence number. We use a variety of tools including SPQR trees and potential maximal cliques, and show that in the bounded cases, one can also obtain tree decompositions with bounded independence number efficiently. This leads to polynomial-time algorithms for the Maximum Weight Independent Set problem in an infinite family of graph classes, each of which properly contains the class of chordal graphs. These results also apply to the class of...

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.

We continue the study of (tw, ω)-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 Maximum Independent Set and related problems.In the previous paper of the series [Dallard, Milanič, and Štorgel, Treewidth versus clique number. II. Tree-independence number], we introduced the tree-independence number, a minmax graph invariant related to tree decompositions. Bounded tree-independence number implies both (tw, ω)-boundedness and the existence of a polynomial-time algorithm for the Maximum Weight Independent Packing problem, provided that the input graph is given together with a tree decomposition with bounded independence number. In particular, this implies polynomial-time solvability of the Maximum Weight Independent Set problem.In this paper, we consider six graph containment relations-the subgraph, topological minor, and minor relations, as well as their induced variants-and for each of them characterize the graphs H for which any graph excluding H 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 K 5 minus an edge or the 4-wheel implies the existence of a tree decomposition in which every bag is a clique plus at most 3 vertices, while excluding a complete bipartite graph K 2,q implies the existence of a tree decomposition with independence number at most 2(q − 1).These results are obtained using a variety of tools, including ℓ-refined tree decompositions, SPQR trees, and potential maximal cliques, and actually show that in the bounded cases identified in this work, one can also compute tree decompositions with bounded independence number efficiently. Applying the algorithmic framework provided by the previous paper in the series leads to polynomial-time algorithms for the Maximum Weight Independent Set problem in an infinite family of graph classes, each of which properly contains the class of chordal graphs. In particular, these results apply to the class of 1-perfectly orientable graphs, answering a question of Beisegel, Chudnovsky, Gurvich, Milanič, and Servatius from 2019.

Functionality is a graph complexity measure that extends a variety of parameters, such as vertex degree, degeneracy, clique-width, or twin-width. In the present paper, we show that functionality is bounded for box intersection graphs in R 1 , i.e. for interval graphs, and unbounded for box intersection graphs in R 3 . We also study a parameter known as symmetric difference, which is intermediate between twin-width and functionality, and show that this parameter is unbounded both for interval graphs and for unit box intersection graphs in R 2 .