arXiv subjects

Sepehr Hajebi

Publications and source records attributed to Sepehr Hajebi.

At least 19 recordsLinked to original sources

arXiv2026

Polynomial bounds for pathwidth

Dallard, Milanič, and Štorgel conjectured that for a hereditary graph class $\mathcal{G}$, if there is some function $f:\mathbb{N}\to\mathbb{N}$ such that every graph $G\in \mathcal{G}$ with clique number $ω(G)$ has treewidth at most $f(ω(G))$, then there is a polynomial function $f$ with the same property. Chudnovsky and Trotignon refuted this conjecture in a strong sense, showing that neither polynomial nor any prescribed growth can be guaranteed in general. Here we prove that, in stark contrast, the analog of the Dallard-Milanič-Štorgel conjecture for pathwidth is true: For every hereditary graph class $\mathcal{G}$, if the pathwidth of every graph in $\mathcal{G}$ is bounded by some function of its clique number, then the pathwidth of every graph in $\mathcal{G}$ is bounded by a polynomial function of its clique number.

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arXiv2026

Forcing monochromatic induced subgraphs

We prove that for all $c\in\mathbb N$ and nonnull graphs $H_1,\ldots,H_t$, there exists $n\in\mathbb N$ such that if $G$ is a $c$-edge-colored complete graph with no monochromatic induced copy of the complete join of $H_1,\ldots,H_t$, then $V(G)$ is the union of $n$ sets $V_1,\ldots,V_n$ such that within each set $V_j$ with $|V_j|\neq 1$, the edges of some color form a graph that excludes at least one of $H_1,\ldots,H_t$ as an induced subgraph. In fact, the same holds even if the colors overlap, and with a different list of graphs $H_1,\ldots,H_t$ assigned to each color. When $H_1,\ldots,H_t$ each have a single vertex, this is Ramsey's theorem, and when $c=2$, this is the "excluding pairs of graphs" theorem of Chudnovsky, Scott, and Seymour.

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arXiv2026

A simple layered-wheel-like construction

In recent years, there has been significant interest in characterizing the induced subgraph obstructions to bounded treewidth and pathwidth. While this has recently been resolved for pathwidth, the case of treewidth remains open, and prior work has reduced the problem to understanding the layered-wheel-like obstructions -- graphs that contain large complete minor models with each branching set inducing a path; exclude large walls as induced minors; exclude large complete bipartite graphs as induced minors; and exclude large complete subgraphs. There are various constructions of such graphs, but they are all rather involved. In this paper, we present a simple construction of layered-wheel-like graphs with arbitrarily large treewidth. Three notable features of our construction are: (a) the vertices of degree at least four can be made to be arbitrarily far apart; (b) the girth can be made to be arbitrarily large; and (c) every outerstring induced subgraph of the graphs from our construction has treewidth bounded by an absolute constant. In contrast, among several previously known constructions of layered wheels, none achieves (a); at most one satisfies either (b) or (c); and none satisfies both (b) and (c) simultaneously. In particular, this is related to a former conjecture of Trotignon, that every graph with large enough treewidth, excluding large walls and large complete bipartite graphs as induced minors, and large complete subgraphs, must contain an outerstring induced subgraph of large treewidth. Our construction provides the first counterexample to this conjecture that can also be made to have arbitrarily large girth.

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arXiv2026

Bull-free graphs and $χ$-boundedness

A bull is a graph obtained from a four-vertex path by adding a vertex adjacent to the two middle vertices of the path. A graph $G$ is bull-free if no induced subgraph of $G$ is a bull. We prove that for all $k,t\in \mathbb N$, if $G$ is a bull-free graph of clique number at most $k$ and every triangle-free induced subgraph of $G$ has chromatic number at most $t$, then $G$ has chromatic number at most $k^{O(\log t)}$. We further show that the bound $k^{O(\log t)}$ is best possible up to a multiplicative constant in the exponent. Thomassé, Trotignon, and Vušković (2017) were the first to give a bound of the form $2^{p\log p}$, where $p=O(k^2+t)$, with a proof that uses Chudnovsky's structure theorem for bull-free graphs. This was improved by Chudnovsky, Cook, Davies, and Oum (2026) to a bound of the form $k^{O(t)}$, with a 10-page proof that again relies heavily on Chudnovsky's structure theorem. Our proof is a single page long and completely avoids the structure theorem, instead using only a result of Chudnovsky and Safra (which itself has a short proof).

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arXiv2026

Asymmetric induced saturation

For which graphs $H$ does there exist a graph $G$ with at least one edge and no induced subgraph isomorphic to $H$, such that deleting any edge of $G$ creates an induced copy of $H$? We call such a graph "$H$-deletion-saturated". This version of the well-studied notion of "$H$-induced-saturated" graphs -- where both adding and deleting any edge creates an induced copy of $H$ -- appears more tractable. For example, while it remains wide open whether $H$-induced-saturated graphs exist for every even cycle $H$, we proved recently that deletion-saturated graphs exist for all even cycles. In fact, apart from complete graphs, no graph $H$ is known for which $H$-deletion-saturated graphs do not exist. We conjecture that $H$-deletion-saturated graphs exist for every non-complete graph $H$, and prove this conjecture for several types of graphs, including: complete bipartite graphs with parts of unequal size, triangle-free graphs with one cycle, graphs with two leaves at distance at most three, and line graphs of trees. In fact, in all cases, we prove the conjecture for substantially more general families. We also verify our conjecture for every graph $H$ on at most six vertices.

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arXiv2026

Tree-alpha and excluding finitely many graphs

We prove that a hereditary graph class $\mathcal{G}$ defined by finitely many excluded induced subgraphs has bounded tree-$α$ if and only if it is "$(\mathrm{tw},ω)$-bounded" (that is, for all $t\in \mathbb N$, the class of all $K_t$-free graphs in $\mathcal{G}$ has bounded treewidth). Equivalently, $\mathcal{G}$ has bounded tree-$α$ if and only if it excludes a complete bipartite graph, a forest whose components each have at most three leaves, and the line graph of such a forest. This resolves two conjectures of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht: the above, and a weaker one that for all $a,b\in \mathbb N$, every hereditary class that excludes $K_{a,a}$ and the $b$-vertex path has bounded tree-$α$. The latter was already open even for $(a,b)\in \{(2,7),(3,5)\}$, and only recently proved for $(a,b)=(2,6)$.

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arXiv2026

Induced subgraphs and tree decompositions XIX. Thetas and forests

Let $H$ be a graph and let $\mathcal{C}$ be a hereditary class of theta-free graphs such that $H\notin \mathcal{C}$. We prove that if (a) $H$ is a forest; and (b) $\mathcal{C}$ excludes the line graphs of all subdivisions of some wall, then the treewidth of every graph in $\mathcal{C}$ is at most a polynomial function of its clique number. This is best possible in that both (a) and (b) are necessary for the existence of $any$ function with the above property.

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arXiv2026

Suns in triangle-free graphs of large chromatic number

For an integer $t\geq 4$, a $t$-sun is a graph obtained from a $t$-vertex cycle $C$ by adding a degree-one neighbor for each vertex of $C$. Trotignon asked whether every triangle-free graph of sufficiently large chromatic number has an induced subgraph that is a $t$-sun for some $t\geq 4$. This remains open, but we show that every triangle-free graph of chromatic number at least $48$ has an induced subgraph that is either a $t$-sun for some $t\geq 5$, or a $4$-sun with a single degree-one vertex deleted. In fact, we prove that for all $\ell\geq 5$, there exists $c=c(\ell)\in \mathbb{N}$ such that every triangle-free graph of chromatic number at least $c$ has an induced subgraph that is either a $t$-sun for some $t\geq \ell$, or a $4$-sun with a single degree-one vertex deleted.

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arXiv2026

Induced subgraphs and tree decompositions XVI. Complete bipartite induced minors

We prove that for every graph $G$ with a sufficiently large complete bipartite induced minor, either $G$ has an induced minor isomorphic to a large wall, or $G$ contains a large constellation; that is, a complete bipartite induced minor model such that on one side of the bipartition, each branch set is a singleton, and on the other side, each branch set induces a path. We further refine this theorem by characterizing the unavoidable induced subgraphs of large constellations as two types of highly structured constellations. These results will be key ingredients in several forthcoming papers of this series.

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arXiv2025

Tree independence number III. Thetas, prisms and stars

We prove that for every $t\in \mathbb{N}$, there exists $τ=τ(t)\in \mathbb{N}$ such that every (theta, prism, $K_{1,t}$)-free graph has tree independence number at most $τ$ (where we allow "prisms" to have one path of length zero).

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arXiv2025

Induced subgraphs and tree decompositions XVII. Anticomplete sets of large treewidth

Two sets $X, Y$ of vertices in a graph $G$ are "anticomplete" if $X\cap Y=\varnothing$ and there is no edge in $G$ with an end in $X$ and an end in $Y$. We prove that every graph $G$ of sufficiently large treewidth contains two anticomplete sets of vertices each inducing a subgraph of large treewidth unless $G$ contains, as an induced subgraph, a highly structured graph of large treewidth that is an obvious counterexample to this statement. These are: complete graphs, complete bipartite graphs and "interrupted $s$-constellations." The latter is a slightly adjusted version of a well-known construction by Bonamy et al.

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arXiv2025

Induced subgraphs and tree decompositions XI. Local structure in even-hole-free graphs of large treewidth

We prove a conjecture of Sintiari and Trotignon that every even-hole-free graph of sufficiently large treewidth contains a four-vertex induced subgraph with at least five edges (that is, either the four-vertex complete graph or the unique four-vertex graph with five edges, also known as the diamond). In fact, we prove two stronger results: (a) For every $K_4$-free chordal graph $H$, every even-hole-free graph of sufficiently large treewidth contains either a four-vertex complete subgraph or an induced subgraph isomorphic to $H$ (when $H$ is the diamond, this yields their conjecture); and (b) For every $K_3$-free chordal graph $H$ (equivalently, for every forest $H$) and every $t \in \mathbb{N}$, every even-hole-free graph of sufficiently large treewidth contains either a $t$-vertex complete subgraph or an induced subgraph obtained from $H$ by adding a universal vertex (when $t=4$ and $H$ is the three-vertex path, this yields their conjecture). The choice of $H$ in both result is best possible: (a) fails for every graph $H$ that is not $K_4$-free and chordal, and (b) fails for every graph $H$ that is not a forest.

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arXiv2025

Tree independence number II. Three-path-configurations

A three-path-configuration is a graph consisting of three pairwise internally-disjoint paths the union of every two of which is an induced cycle of length at least four. A graph is 3PC-free if no induced subgraph of it is a three-path-configuration. We prove that 3PC-free graphs have poly-logarithmic tree-independence number. More explicitly, we show that there exists a constant $c$ such that every $n$-vertex 3PC-free graph graph has a tree decomposition in which every bag has stability number at most $c (\log n)^2$. This implies that the Maximum Weight Independent Set problem, as well as several other natural algorithmic problems, that are known to be NP-hard in general, can be solved in quasi-polynomial time if the input graph is 3PC-free.

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arXiv2025

Halfway to induced saturation for even cycles

For graphs $G$ and $H$, we say that $G$ is $H$-free if no induced subgraph of $G$ is isomorphic to $H$, and that $G$ is $H$-induced-saturated if $G$ is $H$-free but removing or adding any edge in $G$ creates an induced copy of $H$. A full characterization of graphs $H$ for which $H$-induced-saturated graphs exist remains elusive. Even the case where $H$ is a path -- now settled by the collective results of Martin and Smith, Bonamy et al., and Dvoŕǎk -- was already quite challenging. What if $H$ is a cycle? The complete answer for odd cycles was given by Behren et al., leaving the case of even cycles (except for the $4$-cycle) wide open. Our main result is the first step toward closing this gap: We prove that for every even cycle $H$, there is a graph $G$ with at least one edge such that $G$ is $H$-free but removing any edge from $G$ creates an induced copy of $H$ (in fact, we construct $H$-induced-saturated graphs for every even cycle $H$ on at most 10 vertices).

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arXiv2025

Chordal graphs, even-hole-free graphs and sparse obstructions to bounded treewidth

We present and study the following conjecture: for an integer $t\geq 4$ and a graph $H$, every even-hole-free graph of large enough treewidth has an induced subgraph isomorphic to either $K_t$ or $H$, if (and only if) $H$ is a $K_4$-free chordal graph. The ``only if'' part follows from the properties of the so-called layered wheels due to Sintiari and Trotignon. Alecu, Chudnovsky, Spirkl and the author recently proved the conjecture in two special cases: (a) when $t=4$; and (b) when $H=cone (F)$ for some forest $F$; that is, $H$ is obtained from $F$ by adding a universal vertex. Our first result is a common strengthening: for an integer $t\geq 4$ and graphs $F$ and $H$, (even-hole, $cone(cone (F))$, $H$, $K_t$)-free graphs have bounded treewidth if and only if $F$ is a forest and $H$ is a $K_4$-free chordal graph. Also, for general $t\geq 4$, we push the current state of the art further than (b) by settling the conjecture for the smallest choices of $H$ that are not coned forests. This follows from our second result: we prove the conjecture when $H$ is a crystal; that is, a graph obtained from several coned double stars by gluing them together along the middle edges of the double stars. In the first version of this paper, we suggested a strengthening of our main conjecture, that for every $t\geq 1$, every graph of sufficiently large treewidth has an induced subgraph of treewidth $t$ which is either complete, complete bipartite, or $2$-degenerate. This strengthening has now been refuted by Chudnovsky and Trotignon [On treewidth and maximum cliques, arXiv:2405.07471, 2024].

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arXiv2025

Induced subgraphs and tree decompositions X. Towards logarithmic treewidth for even-hole-free graphs

A generalized $t$-pyramid is a graph obtained from a certain kind of tree (a subdivided star or a subdivided cubic caterpillar) and the line graph of a subdivided cubic caterpillar by identifying simplicial vertices. We prove that for every integer $t$ there exists a constant $c(t)$ such that every $n$-vertex even-hole-free graph with no clique of size $t$ and no induced subgraph isomorphic to a generalized $t$-pyramid has treewidth at most $c(t)\log{n}$. This settles a special case of a conjecture of Sintiari and Trotignon; this bound is also best possible for the class. It follows that several \textsf{NP}-hard problems such as \textsc{Stable Set}, \textsc{Vertex Cover}, \textsc{Dominating Set} and \textsc{Coloring} admit polynomial-time algorithms on this class of graphs. Results from this paper are also used in later papers of the series, in particular to solve the full version of the Sintiari-Trotignon conjecture.

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arXiv2025

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

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 $o+2$.

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arXiv2025

Induced subgraphs and tree decompositions XII. Grid theorem for pinched graphs

Given an integer $c\in \mathbb{N}$, we say a graph $G$ is $c$-pinched if $G$ does not contain an induced subgraph consisting of $c$ cycles, all going through a single common vertex and otherwise pairwise disjoint and with no edges between them. What can be said about the structure of $c$-pinched graphs? For instance, $1$-pinched graphs are exactly graphs of treewidth $1$. However, bounded treewidth for $c>1$ is immediately seen to be a false hope because complete graphs, complete bipartite graphs, subdivided walls and line graphs of subdivided walls are all examples of $2$-pinched graphs with arbitrarily large treewidth. There is even a fifth obstruction for larger values of $c$, discovered by Pohoata and later independently by Davies, consisting of $3$-pinched graphs with unbounded treewidth and no large induced subgraph isomorphic to any of the first four obstructions. We fuse the above five examples into a grid-type theorem fully describing the unavoidable induced subgraphs of pinched graphs with large treewidth. More precisely, we prove that for every integer $c\in \mathbb{N}$, a $c$-pinched graph $G$ has large treewidth if and only if $G$ contains one of the following as an induced subgraph: a large complete graph, a large complete bipartite graph, a subdivision of a large wall, the line-graph of a subdivision of a large wall, or a large graph from the Pohoata-Davies construction. Our main result also generalizes to an extension of pinched graphs where the lengths of excluded cycles are lower-bounded.

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