A graph class is \(\chi\)-bounded if the only way to force large chromatic number in graphs from the class is by forming a large clique. In the 1970s, Erdős conjectured that intersection graphs of straight-line segments in the plane are \(\chi\)-bounded, but this was disproved by Pawlik et al. (2014), who showed another way to force large chromatic number in this class\(\unicode{x2014}\)by triangle-free graphs \(B_k\) with \(\chi(B_k) = k\) constructed by Burling (1965). This also disproved the celebrated conjecture of Scott (1997) that classes of graphs excluding induced subdivisions of a fixed graph are \(\chi\)-bounded.

For a fixed integer t ⩾ 1, a (t-)long claw, denoted S_{t,t,t}, is the unique tree with three leaves, each at distance exactly t from the vertex of degree three. Majewski et al. [ICALP 2022, ACM ToCT 2024] proved an analog of the Gyárfás' path argument for S_{t,t,t}-free graphs: given an n-vertex S_{t,t,t}-free graph, one can delete neighborhoods of 𝒪(log n) vertices so that the remainder admits an extended strip decomposition (an appropriate generalization of partition into connected components) into particles of multiplicatively smaller size. In this work, we refine the argument of Majewski et al. to its arguably final form: we show that a constant number of neighborhoods suffice. The statement of Majewski et al. is one of the two pillars of a recent quasi-polynomial time algorithm for Maximum Weight Independent Set in S_{t,t,t}-free graphs [Gartland et al., STOC 2024]; our work immediately improves the quasi-polynomial function in the running time bound. Furthermore, our result significantly simplifies known polynomial-time algorithms for Maximum Weight Independent Set in S_{t,t,t}-free graphs with an additional sparsity assumption such as bounded degree or excluding a fixed biclique as a subgraph.

The hyperbolicity of a graph, informally, measures how close a graph is (metrically) to a tree. Hence, it is intuitively similar to treewidth, but the measures are formally incomparable. Motivated by the broad study of algorithms and separators on planar graphs and their relation to treewidth, we initiate the study of planar graphs of bounded hyperbolicity. Our main technical contribution is a novel balanced separator theorem for planar $δ$-hyperbolic graphs that is substantially stronger than the classic planar separator theorem. For any fixed $δ\geq 0$, we can find balanced separator that induces either a single geodesic (shortest) path or a single geodesic cycle in the graph. An important advantage of our separator is that the union of our separator (vertex set $Z$) with any subset of the connected components of $G - Z$ induces again a planar $δ$-hyperbolic graph, which would not be guaranteed with an arbitrary separator. Our construction runs in near-linear time and guarantees that size of separator is $\mathrm{poly}(δ) \cdot \log n$. As an application of our separator theorem and its strong properties, we obtain two novel approximation schemes on planar $δ$-hyperbolic graphs. We prove that Maximum Independent Set and the Traveling Salesperson problem have a near-linear time FPTAS for any constant $δ$, running in $n\, \mathrm{polylog}(n) \cdot 2^{\mathcal{O}(δ^2)} \cdot \varepsilon^{-\mathcal{O}(δ)}$ time. We also show that our approximation scheme for Maximum Independent Set has essentially the best possible running time under the Exponential Time Hypothesis (ETH). This immediately follows from our third contribution: we prove that Maximum Independent Set has no $n^{o(δ)}$-time algorithm on planar $δ$-hyperbolic graphs, unless ETH fails.

Abstract. We confirm a conjecture of Gartland and Lokshtanov [SODA 2023]: if for a hereditary graph class [Formula: see text] there exists a constant [Formula: see text] such that no member of [Formula: see text] contains a [Formula: see text]-creature as an induced subgraph or a [Formula: see text]-skinny-ladder as an induced minor, then there exists a polynomial [Formula: see text] such that every [Formula: see text] contains at most [Formula: see text] minimal separators. By a result of Fomin, Todinca, and Villanger [ SIAM J. Comput., 44 (2015), pp. 54–87] the latter entails the existence of polynomial-time algorithms for Maximum Weight Independent Set, Feedback Vertex Set and many other problems, when restricted to an input graph from [Formula: see text]. Furthermore, as shown by Gartland and Lokshtanov, our result implies a full dichotomy of hereditary graph classes defined by a finite set of forbidden induced subgraphs into tame (admitting a polynomial bound of the number of minimal separators) and feral (containing infinitely many graphs with exponential number of minimal separators).

We confirm a conjecture of Gartland and Lokshtanov [arXiv:2007.08761]: if for a hereditary graph class 𝒢 there exists a constant k such that no member of 𝒢 contains a k-creature as an induced subgraph or a k-skinny-ladder as an induced minor, then there exists a polynomial p such that every G ∈ 𝒢 contains at most p(|V(G)|) minimal separators. By a result of Fomin, Todinca, and Villanger [SIAM J. Comput. 2015] the latter entails the existence of polynomial-time algorithms for Maximum Weight Independent Set, Feedback Vertex Set and many other problems, when restricted to an input graph from 𝒢. Furthermore, as shown by Gartland and Lokshtanov, our result implies a full dichotomy of hereditary graph classes defined by a finite set of forbidden induced subgraphs into tame (admitting a polynomial bound of the number of minimal separators) and feral (containing infinitely many graphs with exponential number of minimal separators).

We revisit recent developments for the Maximum Weight Independent Set problem in graphs excluding a subdivided claw $S_{t,t,t}$ as an induced subgraph [Chudnovsky, Pilipczuk, Pilipczuk, Thomassé, SODA 2020] and provide a subexponential-time algorithm with improved running time $2^{\mathcal{O}(\sqrt{n}\log n)}$ and a quasipolynomial-time approximation scheme with improved running time $2^{\mathcal{O}(\varepsilon^{-1} \log^{5} n)}$. The Gyárfás' path argument, a powerful tool that is the main building block for many algorithms in $P_t$-free graphs, ensures that given an $n$-vertex $P_t$-free graph, in polynomial time we can find a set $P$ of at most $t-1$ vertices, such that every connected component of $G-N[P]$ has at most $n/2$ vertices. Our main technical contribution is an analog of this result for $S_{t,t,t}$-free graphs: given an $n$-vertex $S_{t,t,t}$-free graph, in polynomial time we can find a set $P$ of $\mathcal{O}(t \log n)$ vertices and an extended strip decomposition (an appropriate analog of the decomposition into connected components) of $G-N[P]$ such that every particle (an appropriate analog of a connected component to recurse on) of the said extended strip decomposition has at most $n/2$ vertices.

A locally surjective homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H) that is surjective in the neighborhood of each vertex in G. In the list locally surjective homomorphism problem, denoted by LLSHom(H), the graph H is fixed and the instance consists of a graph G whose every vertex is equipped with a subset of V(H), called list. We ask for the existence of a locally surjective homomorphism from G to H, where every vertex of G is mapped to a vertex from its list. In this paper, we study the complexity of the LLSHom(H) problem in F-free graphs, i.e., graphs that exclude a fixed graph F as an induced subgraph. We aim to understand for which pairs (H,F) the problem can be solved in subexponential time. We show that for all graphs H, for which the problem is NP-hard in general graphs, it cannot be solved in subexponential time in F-free graphs for F being a bounded-degree forest, unless the ETH fails. The initial study reveals that a natural subfamily of bounded-degree forests F, that might lead to some tractability results, is the family 𝒮 consisting of forests whose every component has at most three leaves. In this case, we exhibit the following dichotomy theorem: besides the cases that are polynomial-time solvable in general graphs, the graphs H ∈ {P₃,C₄} are the only connected ones that allow for a subexponential-time algorithm in F-free graphs for every F ∈ 𝒮 (unless the ETH fails).

Abstract A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines $$\ell _1$$ ℓ 1 and $$\ell _2$$ ℓ 2 , one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, for a given n-vertex graph, whether we can remove at most k vertices to obtain a bipartite permutation graph. This problem is $$\mathsf {NP}$$ NP -complete by the classical result of Lewis and Yannakakis [20]. We analyze the structure of the so-called almost bipartite permutation graphs which may contain holes (large induced cycles) in contrast to bipartite permutation graphs. We exploit the structural properties of the shortest hole in a such graph. We use it to obtain an algorithm for the bipartite permutation vertex deletion problem with running time $${\mathcal {O}}(9^k \cdot n^9)$$ O ( 9 k · n 9 ) , and also give a polynomial-time 9-approximation algorithm.

Let $\Lambda(T)$ denote the set of leaves in a tree $T$. One natural problem is to look for a spanning tree $T$ of a given graph $G$ such that $\Lambda(T)$ is as large as possible. This problem is called maximum leaf number, and it is a well-known NP-hard problem. Equivalently, the same problem can be formulated as the minimum connected dominating set problem, where the task is to find a smallest subset of vertices $D\subseteq V(G)$ such that every vertex of $G$ is in the closed neighborhood of $D$. Throughout recent decades, these two equivalent problems have received considerable attention, ranging from pure graph theoretic questions to practical problems related to the construction of wireless networks. Recently, a similar but stronger notion was defined by Bradshaw, Masařík, and Stacho [ Flexible list colorings in graphs with special degeneracy conditions, in Proceedings of the 31st International Symposium on Algorithms and Computation (ISAAC 2020), LIPIcs. Leibniz Int. Proc. Inform. 181, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2020, article 31]. They introduced a new invariant for a graph $G$, called the robust connectivity and written as $\kappa_\rho(G)$, defined as the minimum value $\frac{|R \cap \Lambda (T)|}{|R|}$ taken over all nonempty subsets $R\subseteq V(G)$, where $T = T(R)$ is a spanning tree on $G$ chosen to maximize $|R \cap \Lambda(T)|$. Large robust connectivity was originally used to show flexible choosability in nonregular graphs. In this paper, we investigate some interesting properties of robust connectivity for graphs embedded in surfaces. We prove a tight asymptotic bound of $\Omega(\gamma^{-\frac{1}{r}})$ for the robust connectivity of $r$-connected graphs of Euler genus $\gamma$. Moreover, we give a surprising connection between the robust connectivity of graphs with an edge-maximal embedding in a surface and the surface connectivity of that surface, which describes to what extent large induced subgraphs of embedded graphs can be cut out from the surface without splitting the surface into multiple parts. For planar graphs, this connection provides an equivalent formulation of a long-standing conjecture of Albertson and Berman [ A conjecture on planar graphs, in Graph Theory and Related Topics, Academic Press, San Diego, CA, 1979, p. 57], which states that every planar graph on $n$ vertices contains an induced forest of size at least $n/2$.

Abstract The 3-coloring of hereditary graph classes has been a deeply-researched problem in the last decade. A hereditary graph class is characterized by a (possibly infinite) list of minimal forbidden induced subgraphs $$H_1,H_2,\ldots $$ H 1 , H 2 , … ; the graphs in the class are called $$(H_1,H_2,\ldots )$$ ( H 1 , H 2 , … ) -free. The complexity of 3-coloring is far from being understood, even for classes defined by a few small forbidden induced subgraphs. For H-free graphs, the complexity is settled for any H on up to seven vertices. There are only two unsolved cases on eight vertices, namely $$2P_4$$ 2 P 4 and $$P_8$$ P 8 . For $$P_8$$ P 8 -free graphs, some partial results are known, but to the best of our knowledge, $$2P_4$$ 2 P 4 -free graphs have not been explored yet. In this paper, we show that the 3-coloring problem is polynomial-time solvable on $$(2P_4,C_5)$$ ( 2 P 4 , C 5 ) -free graphs.

Abstract Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs—a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a richer class of graphs. In particular, mixed unit interval graphs may contain a claw as an induced subgraph, as opposed to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyacı et al. (Inf Process Lett 121:29–33, 2017. 10.1016/j.ipl.2017.01.007 ). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs.

Abstract Let $${\mathcal {C}}$$ C and $${\mathcal {D}}$$ D be hereditary graph classes. Consider the following problem: given a graph $$G\in {\mathcal {D}}$$ G ∈ D , find a largest, in terms of the number of vertices, induced subgraph of G that belongs to $${\mathcal {C}}$$ C . We prove that it can be solved in $$2^{o(n)}$$ 2 o ( n ) time, where n is the number of vertices of G , if the following conditions are satisfied: the graphs in $${\mathcal {C}}$$ C are sparse, i.e., they have linearly many edges in terms of the number of vertices; the graphs in $${\mathcal {D}}$$ D admit balanced separators of size governed by their density, e.g., $${\mathcal {O}}(\varDelta )$$ O ( Δ ) or $${\mathcal {O}}(\sqrt{m})$$ O ( m ) , where $$\varDelta$$ Δ and m denote the maximum degree and the number of edges, respectively; and the considered problem admits a single-exponential fixed-parameter algorithm when parameterized by the treewidth of the input graph. This leads, for example, to the following corollaries for specific classes $${\mathcal {C}}$$ C and $${\mathcal {D}}$$ D : a largest induced forest in a $$P_t$$ P t -free graph can be found in $$2^{\tilde{{\mathcal {O}}}(n^{2/3})}$$ 2 O ~ ( n

Interval graphs, intersection graphs of segments on a real line (intervals), play a key role in the study of algorithms and special structural properties. Unit interval graphs, their proper subclass, where each interval has a unit length, has also been extensively studied. We study mixed unit interval graphs - a generalization of unit interval graphs where each interval has still a unit length, but intervals of more than one type (open, closed, semi-closed) are allowed. This small modification captures a much richer class of graphs. In particular, mixed unit interval graphs are not claw-free, compared to unit interval graphs. Heggernes, Meister, and Papadopoulos defined a representation of unit interval graphs called the bubble model which turned out to be useful in algorithm design. We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of this generalized model by providing a subexponential-time algorithm for solving the MaxCut problem on mixed unit interval graphs. In addition, we derive a polynomial-time algorithm for certain subclasses of mixed unit interval graphs. We point out a substantial mistake in the proof of the polynomiality of the MaxCut problem on unit interval graphs by Boyaci, Ekim, and Shalom (2017). Hence, the time complexity of this problem on unit interval graphs remains open. We further provide a better algorithmic upper-bound on the clique-width of mixed unit interval graphs.

Abstract The k-Colouring problem is to decide if the vertices of a graph can be coloured with at most k colours for a fixed integer k such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list $$L(u)\subseteq \{1,\ldots ,k\},$$ L ( u ) ⊆ { 1 , … , k } , then we obtain the List k-Colouring problem. A graph G is H-free if G does not contain H as an induced subgraph. We continue an extensive study into the complexity of these two problems for H-free graphs. The graph $$P_r+P_s$$ P r + P s is the disjoint union of the r-vertex path $$P_r$$ P r and the s-vertex path $$P_s.$$ P s . We prove that List 3-Colouring is polynomial-time solvable for $$(P_2+P_5)$$ ( P 2 + P 5 ) -free graphs and for $$(P_3+P_4)$$ ( P 3 + P 4 ) -free graphs. Combining our results with known results yields complete complexity classifications of 3-Colouring and List 3-Colouring on H-free graphs for all graphs H up to seven vertices.

The $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for a fixed integer $k$ such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list $L(u) \subseteq \{1,\cdots, k\}$, then we obtain the List $k$-Colouring problem. A graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. We continue an extensive study into the complexity of these two problems for $H$-free graphs. The graph $P_r+P_s$ is the disjoint union of the $r$-vertex path $P_r$ and the $s$-vertex path $P_s$. We prove that List $3$-Colouring is polynomial-time solvable for $(P_2+P_5)$-free graphs and for $(P_3+P_4)$-free graphs. Combining our results with known results yields complete complexity classifications of $3$-Colouring and List $3$-Colouring on $H$-free graphs for all graphs $H$ up to seven vertices.