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Sophie Spirkl

Publications and source records attributed to Sophie Spirkl.

At least 37 records · Page 2Linked to original sources
arXiv2025

The structure of $Δ(1, 2, 2)$-free tournaments

We extend the list of tournaments $S$ for which the complete structural description for tournaments excluding $S$ as a subtournament is known. Specifically, let $Δ(1, 2, 2)$ be a tournament on five vertices obtained from a cyclic triangle by substituting a two-vertex tournament for two of its vertices. In this paper, we show that tournaments excluding $Δ(1, 2, 2)$ as a subtournament are either isomorphic to one of three small tournaments, obtained from a transitive tournament by reversing edges in vertex-disjoint directed paths, or obtained from a smaller tournament with the same property by applying one of two operations. In particular, one of these operations creates a homogeneous set that induces a subtournament isomorphic to one of three fixed tournaments, and the other creates a homogeneous pair such that their union induces a subtournament isomorphic to a fixed tournament. As an application of this result, we present an upper bound for the chromatic number, a lower bound for the size of a largest transitive subtournament, and a lower bound for the number of vertex-disjoint cyclic triangles for such tournaments. The bounds that we present are all best possible.

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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

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

Intersections of graphs and $χ$-boundedness

Given $k$ graphs $G_{1}, \ldots, G_{k}$, their intersection is the graph $(\cap_{i\in [k]}V(G_{i}), \cap_{i\in [k]}E(G_{i}))$. Given $k$ graph classes $\mathcal{G}_{1}, \ldots , \mathcal{G}_{k}$, we call the class $\{G: \forall i \in[k], \exists G_{i} \in \mathcal{G}_{i} \text{ such that } G=G_{1}\cap \ldots \cap G_{k}\}$ the graph-intersection of $\mathcal{G}_{1}, \ldots , \mathcal{G}_{k}$. The main motivation for the work presented in this paper is to try to understand under which conditions graph-intersection preserves $χ$-boundedness. We consider the following two questions: Which graph classes have the property that their graph-intersection with every $χ$-bounded class of graphs is $χ$-bounded? We call such a class intersectionwise $χ$-guarding. We prove that classes of graphs which admit a certain kind of decomposition are intersectionwise $χ$-guarding. We provide necessary conditions that a finite set of graphs $\mathcal{H}$ should satisfy if the class of $\mathcal{H}$-free graphs is intersectionwise $χ$-guarding, and we characterize the intersectionwise $χ$-guarding classes which are defined by a single forbidden induced subgraph. Which graph classes have the property that, for every positive integer $k$, their $k$-fold graph-intersection is $χ$-bounded? We call such a class intersectionwise self-$χ$-guarding. We study intersectionwise self-$χ$-guarding classes which are defined by a single forbidden induced subgraph, and we prove a result which allows us construct intersectionwise self-$χ$-guarding classes from known intersectionwise $χ$-guarding classes.

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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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arXiv2024

Induced subgraphs and tree decompositions XIII. Basic obstructions in $\mathcal{H}$-free graphs for finite $\mathcal{H}$

Unlike minors, the induced subgraph obstructions to bounded treewidth come in a large variety, including, for every $t\geq 1$, the $t$-basic obstructions: the graphs $K_{t+1}$ and $K_{t,t}$, along with the subdivisions of the $t$-by-$t$ wall and their line graphs. But this list is far from complete. The simplest example of a ''non-basic'' obstruction is due to Pohoata and Davies (independently). For every $n \geq 1$, they construct certain graphs of treewidth $n$ and with no $3$-basic obstruction as an induced subgraph, which we call $n$-arrays. Let us say a graph class $\mathcal{G}$ is clean if the only obstructions to bounded treewidth in $\mathcal{G}$ are in fact the basic ones. It follows that a full description of the induced subgraph obstructions to bounded treewidth is equivalent to a characterization of all families $\mathcal{H}$ of graphs for which the class of all $\mathcal{H}$-free graphs is clean (a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to any graph in $\mathcal{H}$). This remains elusive, but there is an immediate necessary condition: if $\mathcal{H}$-free graphs are clean, then there are only finitely many integers $n\geq 1$ such that there is an $n$-array which is $\mathcal{H}$-free. The above necessary condition is not sufficient in general. However, the situation turns out to be different if $\mathcal{H}$ is finite: we prove that for every finite set $\mathcal{H}$ of graphs, the class of all $\mathcal{H}$-free graphs is clean if and only if there is no $\mathcal{H}$-free $n$-array except possibly for finitely many values of $n$.

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arXiv2024

Induced subgraphs and tree decompositions VI. Graphs with 2-cutsets

This paper continues a series of papers investigating the following question: which hereditary graph classes have bounded treewidth? We call a graph $t$-clean if it does not contain as an induced subgraph the complete graph $K_t$, the complete bipartite graph $K_{t, t}$, subdivisions of a $(t \times t)$-wall, and line graphs of subdivisions of a $(t \times t)$-wall. It is known that graphs with bounded treewidth must be $t$-clean for some $t$; however, it is not true that every $t$-clean graph has bounded treewidth. In this paper, we show that three types of cutsets, namely clique cutsets, 2-cutsets, and 1-joins, interact well with treewidth and with each other, so graphs that are decomposable by these cutsets into basic classes of bounded treewidth have bounded treewidth. We apply this result to two hereditary graph classes, the class of ($ISK_4$, wheel)-free graphs and the class of graphs with no cycle with a unique chord. These classes were previously studied and decomposition theorems were obtained for both classes. Our main results are that $t$-clean ($ISK_4$, wheel)-free graphs have bounded treewidth and that $t$-clean graphs with no cycle with a unique chord have bounded treewidth.

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arXiv2024

Orientations of graphs with at most one directed path between every pair of vertices

Given a graph $G$, we say that an orientation $D$ of $G$ is a KT orientation if, for all $u, v \in V(D)$, there is at most one directed path (in any direction) between $u$ and $v$. Graphs that admit such orientations have been used by Kierstead and Trotter (1992), Carbonero, Hompe, Moore, and Spirkl (2023), Briański, Davies, and Walczak (2024), and Girão, Illingworth, Powierski, Savery, Scott, Tamitegami, and Tan (2024) to construct graphs with large chromatic number and small clique number that served as counterexamples to various conjectures. Motivated by this, we consider which graphs admit KT orientations (named after Kierstead and Trotter). In particular, we construct a graph family with small independence number (sublinear in the number of vertices) which admits a KT orientation. We show that the problem of determining whether a given graph admits a KT orientation is NP-complete, even if we restrict ourselves to planar graphs. Finally, we provide an algorithm to decide if a graph with maximum degree at most 3 admits a KT orientation, whereas, for graphs with maximum degree 4, the problem remains NP-complete.

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arXiv2024

Tree Independence Number IV. Even-hole-free Graphs

We prove that the tree independence number of every even-hole-free graph is at most polylogarithmic in its number of vertices. More explicitly, we prove that there exists a constant c>0 such that for every integer n>1 every n-vertex even-hole-free graph has a tree decomposition where each bag has stability (independence) number at most c log^10 n. 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 even-hole-free.

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arXiv2024

Induced subgraphs and tree decompositions XV. Even-hole-free graphs with bounded clique number have logarithmic treewidth

We prove that for every integer $t\geq 1$ there exists an integer $c_t\geq 1$ such that every $n$-vertex even-hole-free graph with no clique of size $t$ has treewidth at most $c_t\log{n}$. This resolves a conjecture of Sintiari and Trotignon, who also proved that the logarithmic bound is asymptotically best possible. 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. As a consequence, for every positive integer $r$, $r$-{\sc Coloring} can be solved in polynomial time on even-hole-free graphs without any assumptions on clique size. As part of the proof, we show that there is an integer $d$ such that every even-hole-free graph has a balanced separator which is contained in the (closed) neighborhood of at most $d$ vertices. This is of independent interest; for instance, it implies the existence of efficient approximation algorithms for certain \textsf{NP}-hard problems while restricted to the class of all even-hole-free graphs.

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arXiv2024

The sandwich problem for odd-hole-free and even-hole-free graphs

For a property $\mathcal{P}$ of graphs, the $\mathcal{P}$-\textsc{Sandwich-Problem}, introduced by Golumbic and Shamir (1993), is the following: Given a pair of graphs $(G_1, G_2)$ on the same vertex set $V$, does there exist a graph $G$ such that $V(G)=V$, $E(G_{1})\subseteq E(G) \subseteq E(G_{2})$, and $G$ satisfies $\mathcal{P}$? A {\em hole} in a graph is an induced subgraph which is a cycle of length at least four. An odd (respectively even) hole is a hole of odd (respectively even) length. Given a class of graphs $\mathcal{C}$ and a graph $G$ we say that $G$ is {\em $\mathcal{C}$-free} if it contains no induced subgraph isomorphic to a member of $\mathcal{C}$. In this paper we prove that if $\mathcal{P}$ is the property of being odd-hole-free or the property of being even-hole-free, then the $\mathcal{P}$-\textsc{Sandwich-Problem} is NP-hard.

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arXiv2024

Graphs of bounded chordality

A hole in a graph is an induced subgraph which is a cycle of length at least four. A graph is chordal if it contains no holes. Following McKee and Scheinerman (1993), we define the chordality of a graph $G$ to be the minimum number of chordal graphs on $V(G)$ such that the intersection of their edge sets is equal to $E(G)$. In this paper we study classes of graphs of bounded chordality. In the 1970s, Buneman, Gavril, and Walter, proved independently that chordal graphs are exactly the intersection graphs of subtrees in trees. We generalize this result by proving that the graphs of chordality at most $k$ are exactly the intersection graphs of convex subgraphs of median graphs of tree-dimension $k$. A hereditary class of graphs $\mathcal{A}$ is $χ$-bounded if there exists a function $ f\colon \mathbb{N}\rightarrow \mathbb{R}$ such that for every graph $G\in \mathcal{A}$, we have $χ(G) \leq f(ω(G))$. In 1960, Asplund and Grünbaum proved that the class of all graphs of boxicity at most two is $χ$-bounded. In his seminal paper "Problems from the world surrounding perfect graphs," Gyárfás (1985), motivated by the above result, asked whether the class of all graphs of chordality at most two, which we denote by $\mathcal{C}\gcap \mathcal{C}$, is $χ$-bounded. We discuss a result of Felsner, Joret, Micek, Trotter and Wiechert (2017), concerning tree-decompositions of Burling graphs, which implies an answer to Gyárfás' question in the negative. We prove that two natural families of subclasses of $\mathcal{C}\gcap \mathcal{C}$ are polynomially $χ$-bounded. Finally, we prove that for every $k\geq 3$ the $k$-\textsc{Chordality Problem}, which asks to decide whether a graph has chordality at most $k$, is \textbf{NP}-complete.

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arXiv2024

List-3-Coloring ordered graphs with a forbidden induced subgraph

The List-3-Coloring Problem is to decide, given a graph $G$ and a list $L(v)\subseteq \{1,2,3\}$ of colors assigned to each vertex $v$ of $G$, whether $G$ admits a proper coloring $ϕ$ with $ϕ(v)\in L(v)$ for every vertex $v$ of $G$, and the $3$-Coloring Problem is the List-$3$-Coloring Problem on instances with $L(v)=\{1,2,3\}$ for every vertex $v$ of $G$. The List-$3$-Coloring Problem is a classical NP-complete problem, and it is well-known that while restricted to $H$-free graphs (meaning graphs with no induced subgraph isomorphic to a fixed graph $H$), it remains NP-complete unless $H$ is isomorphic to an induced subgraph of a path. However, the current state of art is far from proving this to be sufficient for a polynomial time algorithm; in fact, the complexity of the $3$-Coloring Problem on $P_8$-free graphs (where $P_8$ denotes the eight-vertex path) is unknown. Here we consider a variant of the List-$3$-Coloring Problem called the Ordered Graph List-$3$-Coloring Problem, where the input is an ordered graph, that is, a graph along with a linear order on its vertex set. For ordered graphs $G$ and $H$, we say $G$ is $H$-free if $H$ is not isomorphic to an induced subgraph of $G$ with the isomorphism preserving the linear order. We prove, assuming $H$ to be an ordered graph, a nearly complete dichotomy for the Ordered Graph List-$3$-Coloring Problem restricted to $H$-free ordered graphs. In particular, we show that the problem can be solved in polynomial time if $H$ has at most one edge, and remains NP-complete if $H$ has at least three edges. Moreover, in the case where $H$ has exactly two edges, we give a complete dichotomy when the two edges of $H$ share an end, and prove several NP-completeness results when the two edges of $H$ do not share an end, narrowing the open cases down to three very special types of two-edge ordered graphs.

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arXiv2024

Tree independence number I. (Even hole, diamond, pyramid)-free graphs

The tree-independence number tree-$α$, first defined and studied by Dallard, Milanič and Štorgel, is a variant of treewidth tailored to solving the maximum independent set problem. Over a series of papers, Abrishami et al. developed the so-called central bag method to study induced obstructions to bounded treewidth. Among others, they showed that, in a certain superclass $\mathcal C$ of (even hole, diamond, pyramid)-free graphs, treewidth is bounded by a function of the clique number. In this paper, we relax the bounded clique number assumption, and show that $\mathcal C$ has bounded tree-$α$. Via existing results, this yields a polynomial time algorithm for the maximum independent set problem in this class. Our result also corroborates, for this class of graphs, a conjecture of Dallard, Milanič and Štorgel that in a hereditary graph class, tree-$α$ is bounded if and only if the treewidth is bounded by a function of the clique number.

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