Induced subgraphs and tree decompositions IX. Grid theorem for perforated graphs

TL;DR

This work establishes a grid-type theorem for induced subgraphs in the class of c-perforated graphs, showing that large treewidth guarantees the presence of either a large clique, a large complete bipartite graph, or a large full occultation (and extends the framework to (c,o)-perforated graphs with girth constraints). It introduces and systematizes asterisms, necklaces of paths, and the novel notions of contrived obstructions (occultations and full occultations) that play the role of non-basic obstructions in inducing grid-like structure. The proof develops a multi-layered machinery including constellations, transition graphs, patch & match constructs, and a sequence of Ramsey-type arguments to move from high connectivity to concrete induced subgraph obstructions. The results advance the understanding of how induced-subgraph forbiddance interacts with treewidth, delivering the first grid-type theorem for a hereditary class that uses non-basic obstructions beyond subdivision walls and their line graphs, with implications for structure theory and algorithmic applications in sparse and structured graph classes.

Abstract

The celebrated Erdős-Pósa Theorem, in one formulation, asserts that for every $c\geq 1$, graphs with no subgraph (or equivalently, minor) isomorphic to the disjoint union of $c$ cycles have bounded treewidth. What can we say about the treewidth of graphs containing no induced subgraph isomorphic to the disjoint union of $c$ cycles? Let us call these graphs $c$-perforated. While $1$-perforated graphs have treewidth one, complete graphs and complete bipartite graphs are examples of $2$-perforated graphs with arbitrarily large treewidth. But there are sparse examples, too: Bonamy, Bonnet, Déprés, Esperet, Geniet, Hilaire, Thomassé and Wesolek constructed $2$-perforated graphs with arbitrarily large treewidth and no induced subgraph isomorphic to $K_3$ or $K_{3,3}$; we call these graphs occultations. Indeed, it turns out that a mild (and inevitable) adjustment of occultations provides examples of $2$-perforated graphs with arbitrarily large treewidth and arbitrarily large girth, which we refer to as full occultations. Our main result shows that the converse also holds: for every $c\geq 1$, a $c$-perforated graph has large treewidth if and only if it contains, as an induced subgraph, either a large complete graph, or a large complete bipartite graph, or a large full occultation. This distinguishes $c$-perforated graphs, among graph classes purely defined by forbidden induced subgraphs, as the first to admit a grid-type theorem incorporating obstructions other than subdivided walls and their line graphs. More generally, for all $c,o\geq 1$, we establish a full characterization of induced subgraph obstructions to bounded treewidth in graphs containing no induced subgraph isomorphic to the disjoint union of $c$ cycles, each of length at least $o+2$.

Figures (15)

• Figure 1: The graph $W_{5 \times 5}$.
• Figure 2: A $4$-occultation where $\pi(i)=x_i$ for $i=1,2,3,4$.
• Figure 3: A full $4$-occultation with $\pi(i)=x_i$ for $i=1,2,3,4$.
• Figure 4: Outcomes of Theorem \ref{['connectifier']} when $h=5$. From left to right: the first for \ref{['thm:minimalconnectedgeneral_a']}, the second for \ref{['thm:minimalconnectedgeneral_b']} when $r\notin S$, the third for \ref{['thm:minimalconnectedgeneral_b']} when $r\in S$, the fourth for \ref{['thm:minimalconnectedgeneral_c']}, and the fifth and the sixth for \ref{['thm:minimalconnectedgeneral_d']}.
• Figure 5: A $5$-asterism $\mathfrak{a}$ with $S_{\mathfrak{a}}=\{x_1,\ldots, x_5\}$ and $L_{\mathfrak{a}}=v_1\hbox{-} \cdots \hbox{-} v_{25}$. Note, for instance, that $x_2\hbox{-} v_{13}\hbox{-} v_{14}\hbox{-} v_{15} \hbox{-} v_{16}\hbox{-} x_4$ is an $\mathfrak{a}$-route that is not minimal, and $x_2\hbox{-} v_{13}\hbox{-} v_{14}\hbox{-} x_3$ is an $\mathfrak{a}$-route that is minimal. Also, there are exactly 15 $\mathfrak{a}$-pieces; 13 of which internal and $v_1\hbox{-} v_{2}\hbox{-} v_{3}$ and $v_{22}\hbox{-} v_{23}\hbox{-} v_{24}\hbox{-} v_{25}$ are the two external $\mathfrak{a}$-pieces. For instance, $v_{17}\hbox{-} v_{18}\hbox{-} v_{19}$ is an internal $\mathfrak{a}$-piece which is open, and $v_{5}\hbox{-} v_{6}\hbox{-} v_{7}\hbox{-} v_{8}$ is a closed $\mathfrak{a}$-piece (which is necessarily internal).

Theorems & Definitions (47)

• Theorem 1.1: Robertson and Seymour RS-GMV
• Theorem 1.2: Abrishami, Alecu, Chudnovsky, Hajebi and Spirkl twvii
• Theorem 1.3: Erdős and Pósa ErdosPosa
• Theorem 1.4
• Theorem 1.5: Bonamy, Bonnet, Déprés, Esperet, Geniet, Hilaire, Thomassé and Wesolek deathstar
• Theorem 1.6: Bonamy, Bonnet, Déprés, Esperet, Geniet, Hilaire, Thomassé and Wesolek deathstar
• Theorem 2.1: Ramsey multiramsey
• Theorem 2.2: Graham, Rothschild and Spencer productramsey, see also KP
• Theorem 2.3: Dvořák, see Theorem 5 in dvorak, Lozin and Razgon, see Theorem 3 in lozin
• Theorem 2.4: Abrishami, Alecu, Chudnovsky, Hajebi and Spirkl twvii