graph algorithm designers

Abstract: One way to compare two graphs $G_1, G_2$ is to check whether there is some graph $H$ in a graph class $\mathcal{H}$, such that the number of homomorphisms from $H$ to $G_1$ is not equal to that from $H$ to $G_2$. This is the field of homomorphism distinguishability which has gained a lot of traction in recent years. In this paper, we study the case where $\mathcal{H}$ is the class of graphs of treewidth $k$. This has connections to important concepts in logic, machine learning, combinatorial optimization and proof complexity. In particular, Dvořák showed that $G_1$ and $G_2$ can be distinguished by a homomorphism count from a graph of treewidth $k$ iff $G_1$ and $G_2$ are distinguishable by the $k$-dimensional Weisfeiler-Leman algorithm. While the relation between the iteration number of this algorithm and the depth of the tree-decomposition of a distinguishing graph is understood, our understanding of the \emph{size} of $H$ is limited. We show a $2^{Ω(n^{k/2})}$ lower bound on the size of a graph of treewidth $k$ required to distinguish two graphs of order $n$. This bound is proved via a new combinatorial game along with a construction which uses a recent compression technique for CFI graphs. It is almost tight to the known $2^{O(n^{k+1})}$ upper bound and is an exponential improvement on the previous best known lower bound.

Abstract: We prove that every claw-free perfect graph $G$ satisfies $\ch(G)=χ(G)=ω(G)$. More strongly, if $S$ is a clique of simplicial vertices of $G$ and $k\geω(G)$, every proper precoloring of $S$ extends to a proper coloring of $G$ from arbitrary lists of size $k$ on $V(G)\setminus S$. The proof has three coloring steps: a mixed-list theorem for cobipartite graphs, a precoloring-extension theorem for elementary graphs built on Galvin's kernel method and a homogeneous-pair exchange argument, and a polynomial method (Combinatorial Nullstellensatz) argument for graphs of interval type in which a Vandermonde factor is transferred across each incomplete pair. The three coloring steps are assembled along the structure theorems of Chvátal--Sbihi, Maffray--Reed and Chudnovsky--Seymour, using double clique cutsets for composition. As a consequence, every perfect line graph is chromatic-choosable, recovering the theorem of Peterson and Woodall. Our precoloring extension theorem also resolves a conjecture posed by Chudnovsky and Seymour.

Abstract: We prove that finite two-graphs have coherent EPPA and that neither graphs modulo global complementation nor two-graphs modulo global complementation have EPPA. This completes the classification of EPPA and coherent EPPA for the reducts of the random graph, answering a question and confirming a conjecture of Evans, Hubička, Konečný and Nešetřil.

Abstract: Comparability graphs represent partial orders and provide natural building blocks for representing a graph as a union of simpler graphs. The analogous task in the directed setting is to represent a digraph as a union of partial orders. Aboulker et al. [arXiv, 2026] recently introduced the partial order decomposition number as a useful structural parameter in their study of tournaments. Formally, a partial order decomposition of a digraph $D$ is a cover of its arc set by transitive acyclic subdigraphs, each representing a strict partial order on the vertex set of $D$. The partial order decomposition number is the minimum number of such subdigraphs needed. We also consider the partitioning variant, in which every arc belongs to exactly one of the partial orders in the decomposition. We prove that, for every fixed $k\geq 2$, deciding whether a digraph admits a decomposition into at most $k$ partial orders is NP-complete, even for directed acyclic graphs (DAGs). For $k=2$, we reduce from NAE-3-SAT. For $k\geq 3$, we reduce from $k$-Coloring, and the reduction yields DAGs with triangle-free underlying graphs. Both reductions establish hardness for both variants. We further show that the minimum cover size and the minimum partition size can differ even on DAGs.

Abstract: If $G$ and $H$ are two cubic graphs, then an $H$-coloring of $G$ is a mapping $f: E(G)\rightarrow E(H)$, such that for every $v\in V(G)$ there is a vertex $x\in V(H)$, such that $f(\partial_G(v))=\partial_H(x)$. If $G$ admits an $H$-coloring then it is common to write $H\prec G$. In this paper, for a fixed connected cubic graph $H$, we consider a decision problem, which we call $H$-coloring. In this problem, on the input one is given a connected cubic graph $G$. The goal is to decide whether $H\prec G$. A conjecture by Hakobyan and the author states that $H$-coloring is polynomial time solvable, if $H$ does not admit a $P_{12}$-coloring. Here $P_{12}$ is the unique cubic graph obtained from the Petersen graph by expanding one of its vertices into a triangle. In this paper, we disprove this conjecture under $P\neq NP$, by showing that $S_6$-coloring is NP-complete. Here $S_6$ is the unique cubic graph on six vertices having exactly one bridge. Our second result is an algorithm that solves $H$-coloring problem in time $O^*(c^n)$, where $H$ is a simple connected cubic graph, $n$ is the number of vertices of the input cubic graph $G$ and $c=\sqrt{8}\approx 2.8284$.

Abstract: We prove Wang's planar D-coloring conjecture. A D-coloring of a graph is a proper edge coloring in which every subgraph isomorphic to $K_4$ minus an edge is rainbow. Wang conjectured that every planar graph of maximum degree $Δ$ has a D-coloring with 9 colors if $Δ=4$, 10 colors if $Δ=5$, and $2Δ-1$ colors if $Δ\ge 6$. Hu, Kong and Wang proved the cases $Δ\le 5$ and $Δ\ge 33$. We prove the sharp bound $2Δ-1$ for every $Δ\ge 6$, settling the remaining cases. The proof uses discharging; the case $Δ=6$ is computer-assisted.

Abstract: A \emph{proper vertex coloring} assigns colors to vertices so that adjacent vertices have distinct colors. A \emph{Kempe change} in a proper vertex coloring exchanges two colors on one component of the subgraph induced by the vertices of those colors. For a fixed palette of $k$ colors, \textsc{Kempe Reachability} asks whether two given proper vertex colorings of the same graph are connected by a sequence of Kempe changes. This problem is known to be $\mathsf{PSPACE}$-complete for every fixed $k\geq 3$. A graph is \emph{claw-free} if no vertex has three pairwise nonadjacent neighbors. The \emph{line graph} of a graph has one vertex for each edge, with two vertices adjacent exactly when the corresponding edges share an end. Line graphs are claw-free, and their Kempe changes correspond to \emph{edge-Kempe changes} in the original graph, which exchange two colors on one component of the subgraph formed by the edges of those colors. A claw-free graph with a proper $k$-coloring has maximum degree at most $2k-2$, whereas for $k\geq 4$ all proper $k$-colorings of a graph of maximum degree at most $k$ are connected by Kempe changes. We show that, for every fixed $k\geq 3$, \textsc{Kempe Reachability} with $k$ colors is $\mathsf{PSPACE}$-complete even on line graphs of connected simple $k$-regular graphs, and hence even on connected $(2k-2)$-regular claw-free graphs. Equivalently, deciding whether two proper $k$-edge-colorings are connected by edge-Kempe changes is $\mathsf{PSPACE}$-complete even on connected simple $k$-regular graphs. ChatGPT 6 Pro and Claude Fable 5.1 were used to assist with developing and reviewing mathematical arguments and drafting and revising the paper. The author takes full responsibility for the correctness and content of the paper.

Abstract: For a partition $\boldsymbolλ = (λ_1,\ldots,λ_r)$ and $n\ge r$, the monomial symmetric polynomial is defined as $$m_{\boldsymbolλ}(x_1,\ldots,x_n) := \sum x_{i_1}^{λ_1} x_{i_2}^{λ_2} \cdots x_{i_r}^{λ_r},$$ where the sum ranges over all distinct monomials obtained by assigning the parts of $\boldsymbolλ$ to pairwise distinct variables. We establish an almost-complete dichotomy for the complexity of the monomial symmetric polynomials $m_{\boldsymbolλ}$: If a family of partitions contains polynomially many distinct parts, then the corresponding monomial symmetric polynomials (on sufficiently many variables) are VNP-complete. If the number of distinct parts is constant, the family admits polynomial-size circuits. Thus the only remaining gap concerns the intermediate regime where the number of distinct parts grows subpolynomially. As part of our proof, we use combinatorial properties of matching polynomials to establish VNP-hardness of the perfect matching polynomial for blow-up graphs; these are graphs with non-generic edge-weights that are obtained by "blowing up" a smaller graph. We reduce these matching polynomials to monomial-symmetric polynomials by combining classical techniques from algebraic complexity theory with structural properties of Sidon sets.

Abstract: We study the fixed-parameter tractability of the \textsc{Isometric Path Cover} problem where, given an integer $k$ and a simple, undirected, and unweighted graph $G$, the objective is to decide whether $G$ admits a set of $k$ isometric paths (i.e., shortest paths between their endpoints) whose union is the entire vertex set of $G$. We show that \textsc{Isometric Path Cover} is NP-hard on graphs of pathwidth (and thus treewidth) at most 287. This is an interesting contrast with the closely related problems \textsc{Isometric Path Partition} and \textsc{Path Cover}, which are both known to be XP for treewidth, see [Chakraborty et al., WG 2026] and [Foucaud et al., TCS 2026]. On the positive side, we show, by a suitable dynamic programming scheme, that \textsc{Isometric Path Cover} admits an XP algorithm w.r.t.~the parameter combination of treewidth and diameter, running in time $n^{\diam^{O(\tw)}}$ where $\diam$, $\tw$, and $n$ denote the diameter, the treewidth, and the number of vertices of the input graph, respectively. However, we also prove that \textsc{Isometric Path Cover} becomes W[1]-hard for pathwidth on graphs of diameter~7, showing that an XP algorithm is the best one can hope for w.r.t.~this parameterization, under standard complexity theory assumptions.

Abstract: The Eccentricity Shortest Path problem asks, given a connected graph and a nonnegative integer $R$, whether some shortest path is within distance $R$ of every vertex. We study its structural parameterized complexity. We give fixed-parameter algorithms for vertex-deletion distance to cographs, vertex-deletion distance to interval graphs, and bandwidth. The interval-deletion result answers a question of Bhyravarapu et al. (IWOCA 2023). We obtain XP algorithms for vertex-deletion distance to chordal graphs, clique-width, and rank-width, without requiring expressions or decompositions for the latter two. Consequently, the problem is polynomial-time solvable for every fixed treewidth, answering a question of Chakraborty et al. (2026). We also derive fixed-parameter tractability for treedepth and XP membership for diameter from earlier algorithms. We prove $\mathsf{W}[3]$-hardness for cutwidth, and hence pathwidth and treewidth, even on graphs of maximum degree at most five; $\mathsf{W}[2]$-hardness for vertex clique-cover number; and NP-hardness on graphs of tree-independence number at most two. For maximum leaf number $\ell$, we obtain an $O(\ell^4)$-bit compression and a polynomial kernel. Finally, unless $\mathsf{NP}\subseteq\mathrm{coNP}/\mathrm{poly}$, we exclude polynomial kernels for bandwidth, feedback edge number, vertex-deletion distance to a clique, and vertex cover number. The last bound also excludes polynomial kernels for vertex-deletion distance to proper interval graphs.

Abstract: Consider a directed graph G and the (generally infinite) set Walks(G) of the walks in that graph. We are interested in ordering Walks(G) as w1, w2,... in a way that ensures that each walk is obtained from the previous one by a bounded number of elementary operations, namely, push operations (which add a new edge at either end of the walk) and pop operations (which remove the first or last edge of the walk). Our goal is to optimize the aforementioned bound, when it exists, in particular when imposing the additional requirement that walks must be ordered in increasing order of length. To this end, we introduce a condition on graphs inspired by earlier work, called inner-connectedness, which is testable in linear time; we show that inner-connected graphs are precisely those which admit bounded walk orderings. Focusing on such graphs, we then introduce and study the pp-rank and the pp-increasing-rank. Namely, the pp-rank of a graph is the best possible bound on the number of push-pop operations in a bounded walk ordering for that graph; and the pp-increasing-rank is the same where we further impose that shorter walks appear before longer walks in the ordering. We show that the pp-rank of inner-connected graphs is in fact always bounded by 3. The pp-increasing-rank of inner-connected graphs is also always finite, but it can be up to linear in the input graph. More precisely, we show that the pp-increasing-rank is related to a new parameter we introduce, the peripherality of graphs. In particular, for cyclic graphs, peripherality is related to the pp-increasing-rank up to multiplicative and additive constants. We last study how to compute these parameters, and show that while peripherality can be computed in polynomial time, the pp-increasing-rank itself is generally NP-hard to compute.

Abstract: For Vertex Cover, a minimal blocking set of a graph $G$ is an inclusion-wise minimal vertex set contained in no minimum vertex cover of $G$. Bougeret et al. [SIDMA 2022] showed that, for all minor-closed graph classes $\mathcal{G}$, Vertex Cover has a polynomial kernel parameterized by the distance to $\mathcal{G}$ if and only if $\mathcal{G}$ has bounded minimal blocking sets. In this work, we explore how far this connection between minimal blocking sets and kernelization extends to $\mathcal{F}$-Minor Deletion parameterized by the deletion distance to $\mathcal{G}$. A minimal $\mathcal{F}$-blocking set of $G$ is a minimal vertex set not contained in any optimal $\mathcal{F}$-Minor Deletion solution of $G$. For connected families $\mathcal{F}$, we show that if some graph class $\mathcal{G}$ closed under disjoint union has unbounded minimal $\mathcal{F}$-blocking sets, then $\mathcal{F}$-Minor Deletion parameterized by the deletion distance to $\mathcal{G}$ has no polynomial kernel. This generalizes a result of Hols et al. [SIDMA 2022] for Vertex Cover. We then show that this criterion does not capture all kernelization lower bounds for $\mathcal{F}$-Minor Deletion, even for minor-closed graph classes. Motivated by the work of Dekker and Jansen [Discrete Appl. Math. 2024], we study the more general notion of extended $\mathcal{F}$-blocking sets and the kernelization lower bounds they yield. For every fixed integer $η\geq 1$, Pw-$η$ Deletion is the problem of deleting at most $k$ vertices to reduce the pathwidth to at most $η$. We show that it has no polynomial kernel parameterized by the distance to graphs with pathwidth at most $η+1$. Moreover, for connected families $\mathcal{F}$, we show that $\mathcal{F}$-Minor Deletion parameterized by the distance to graphs with pathwidth at most $\max_{F\in\mathcal{F}} pw(F)+1$ has no polynomial kernel.

Abstract: We construct trees with perfect matchings and independence sequences that fail log-concavity near the independence number $α$. For every integer $k\geq1$, one family fails at $α-k$ and has a witness of order $O(k^2\log(k+2))$. For every integer $t\geq2$, a single tree has simultaneous failures at $α-1,α-3,\ldots,α-(2t-1)$, with a witness of order $O(t^3)$. All these trees satisfy $α=N/2$, where $N$ is the number of vertices, and contradict Conjecture 4.2 of Ramos and Sun. A sufficient condition uses a fixed tree and the numbers of attached leaves. For fixed $k$ or $t$, explicit bounds give geometric decay of $a_j^2/(a_{j-1}a_{j+1})$ at these indices as the trees grow; $a_j$ counts independent $j$-sets. The proofs count independent sets by empty matching pairs. The Lean proof assistant checks formal proofs of the parameterized results and their quantitative bounds; exact 90- and 180-vertex examples illustrate the constructions.

Abstract: Fix a set $S$ of positive integers. An $S$-Hamiltonian cycle is a cyclic ordering of all vertices of a graph such that each consecutive pair is joined by a walk whose length belongs to $S$. We study the complexity of deciding whether an input graph admits such an ordering, together with the corresponding path variants with prescribed or unrestricted endpoints. We complete the complexity classification for undirected graphs. In particular, we prove NP-completeness for every finite nonempty set of odd lengths, resolving an open problem posed by Amarilli, Lombardo, and Monet (WG 2026). For digraphs, all three problems are NP-complete for every fixed finite nonempty $S$ and admit linear-time algorithms when $S$ is cofinite. For other fixed infinite sets $S$, we establish NP-complete and undecidable cases, with undecidability occurring even when $S$ is recursive.

Abstract: For a fixed finite collection of graphs $\mathcal{F}$, the $\mathcal{F}$-Minor-Free Deletion problem asks, given a graph $G$ and an integer $k$, whether there is a set $S\subseteq V(G)$ with $|S|\leq k$ such that $G-S$ contains no graph of $\mathcal{F}$ as a minor. This generalizes Vertex Cover, Feedback Vertex Set, and Vertex Planarization, among many others. Baste, Sau, and Thilikos [JCSS 2020, SICOMP 2023] showed that, assuming the Exponential Time Hypothesis (ETH), there are collections $\mathcal{F}$ for which the problem cannot be solved in time $2^{o(\mathrm{tw}\log\mathrm{tw})}n^{O(1)}$ on $n$-vertex graphs of treewidth $\mathrm{tw}$, and gave a matching algorithm. We ask whether this super-exponential dependence persists under more restrictive structural parameters. First, we strengthen their lower bound. Assuming ETH, for every fixed $h\geq 6$, $\{P_h\}$-Minor-Free Deletion, where $P_h$ is the path on $h$ vertices, cannot be solved in time $2^{o(t\log t)}n^{O(1)}$, where $t=\mathrm{fvs}+\mathrm{td}$ is the sum of the feedback vertex set number and treedepth of the input graph. These two parameters are incomparable, but both individually upper bound the treewidth. Together with the $2^{O(\mathrm{tw}\log\mathrm{tw})}n^{O(1)}$-time algorithm of Baste, Sau, and Thilikos [SICOMP 2023], our lower bound is tight. In contrast, we show that asymptotically faster algorithms are possible when parameterized by the vertex cover number $\mathrm{vc}$. Specifically, for any finite family $\mathcal{F}$, $\mathcal{F}$-Minor-Free Deletion can be solved in time $2^{O(\mathrm{vc})}n^{O(1)}$. Taken together, our results show that the boundary between single-exponential and super-exponential complexity for this problem lies between the vertex cover number and the feedback vertex set number (plus treedepth).

Abstract: The central object in this work is the Boolean network, a function $f:\{0,1\}^V \to \{0,1\}^V$. Often identified with $f$ is its dynamics $D_f$, the digraph with exactly $\{x \to y : f(x)=y\}$ as its arcs. The interaction graph of $f$ is the digraph $G$ on $V$ which can be thought of as representing the dependencies of the components of $f$ on one another. On the other hand, model-checking is a major field of study in graph algorithms and complexity. We consider problems at the intersection of these paradigms: for a given property $φ$, we say that a digraph $G$ is $φ$-dynamic if $G$ is the interaction graph of some Boolean network $f$ such that $D_f \models φ$. The corresponding decision problem $φ$-DYNAMIC asks whether $G$ is $φ$-dynamic. Problems of this form are well-studied in the Boolean networks community. For example, it is a well-known result that, letting $φ$ be 'No fixed points' (i.e. $\nexists x:x\to x$), $φ$-DYNAMIC is equivalent to the structural digraph problem NON-ACYCLIC, which is solvable in linear time. We establish, for the first time, properties $φ$ for which $φ$-DYNAMIC is computationally hard. These include the 'Unique fixed point' property $\exists!x:x\to x$, and a range of properties concerning the periodic points of $f$ (i.e. vertices of $D_f$ which lie on a cycle). More widely, we exhibit an infinite family of natural properties $φ$ such that $φ$-DYNAMIC is $\mathrm{NEXPTIME}$-complete, which we call PERIOD PIECE. As a special case, these include all properties of the form 'Unique cycle of length $k$' and 'Exactly $k$ fixed points' for any $k \ge 1$. On the other hand, we show that if the input digraph has bounded in-degree, then for those same problems $φ$-DYNAMIC collapses down to $\mathrm{NP}^{\mathrm{NP}}$, the second level of the polynomial hierarchy, though it remains $\mathrm{NP}^{\mathrm{NP}}$-complete.

Abstract: Conway's $99$-graph is a strongly regular graph with parameters $(v,k,λ,μ) = (99,14,1,2)$ and whose existence has long been an open problem. We use both a sum-of-squares program and the fact that a strongly regular graph with $λ= 1, μ= 2$ has no diamond or $K_{2,3}$ subgraph to derive a loose upper bound for the number of triangles that such a graph could contain. This bound is less than the number of triangles that the graph's spectrum asserts, implying that it does not exist.

Abstract: A key milestone in algorithmic graph theory is Hopcroft and Tarjan's 1974 linear-time algorithm for graph planarity testing. At a workshop in 2002, Linial and Rabinovich asked for either a polynomial-time algorithm or an NP-hardness proof for metric planarity testing: given a metric space $(X,d_X)$, determine whether there exists an edge-weighted planar graph $G=(V,E,w)$ with $X\subseteq V$ that realizes the distances in $X$, meaning that $d_G(x,y)=d_X(x,y)$ for every $x,y\in X$. This question arose in the context of research on metric embeddings and their applications in computer science since the mid-1990s. Over the past few decades, polynomial-time algorithms have been developed for several special cases. In this work, we show that metric planarity testing is NP-hard, in sharp contrast to graph planarity testing. Our reduction yields a stronger result: under the Exponential Time Hypothesis (ETH), no $2^{o(n)}$-time algorithm for metric planarity testing exists, where $n$ is the number of points in the metric. To complement our ETH lower bound, we give two algorithmic results: (i) an algorithm for computing a metric planar embedding in $2^{\tilde{O}(n^2)}$ time and (ii) a $(1+ε)$-approximation algorithm in $2^{\tilde{O}(n)}$ time for any fixed $ε\in(0,1)$. The second algorithm produces a planar graph that preserves the input metric up to a factor of $(1+ε)$.

Abstract: We prove that for all $\varepsilon>0$, there exists a positive integer $k$ such that given a $4$-to-$1$ game $Ψ$ with alphabet size at most $k$, it is $\mathbf{NP}$-hard to distinguish between the case that $\mathrm{val}(Ψ)=1$ and the case that $\mathrm{val}(Ψ)\leq \varepsilon$. This confirms the $4$-to-$1$ Games Conjecture from [Khot, \textit{CCC 2002}]. Previously, the best known result, due to [Dinur, Khot, Kindler, Minzer, Safra], established the almost-perfect completeness version (but applied to the stricter problem of $2$-to-$1$ games). Using results from the literature, we get the following implications: (1) for all $k\in \mathbb{N}$, given a $3$-colorable graph $G$, it is $\mathbf{NP}$-hard to find a proper $k$-coloring; (2) for all $δ>0$, given a $2$-colorable $3$-uniform hypergraph $G$, it is $\mathbf{NP}$-hard to find in it an independent set containing at least $δ$ fraction of the vertices. Our proof is a three-step construction that builds on the two-step framework of [Dinur, Khot, Kindler, Minzer, Safra]. In the outer-PCP step, we use quadratic equations to gain perfect completeness. We then construct a new middle PCP that performs low-rank tests while preserving a key covering property. Finally, we construct a new inner PCP based on a tensor of the standard Grassmann encoding with its low-rank variant due to [Golowich, FOCS 2023].

Abstract: Addario-Berry, Reed, Scott, and Wood (JoCG 2026) proved that the $n$-dimensional associahedron has chromatic number at most $22500\cdot\log_3 n+O(1)$. We present a much simpler proof that the chromatic number of the $n-1$-dimensional associahedron is at most $6 \log_2 n + 18$. Since this paper's appearance as a preliminary abstract in the Japan Conference on Discrete and Computational Geometry, Graphs, and Games held from September 7-10, 2026, the bound has been improved to $O(\log \log n)$ by Oum and Wood (arXiv September 30, 2026). It is hoped that the simple method here could give insights to further improvements.

Abstract: The Weisfeiler-Leman (WL) algorithm tests graph isomorphism by assigning colours to $d$-tuples of vertices and iteratively refining the colouring according to neighbourhood configurations until it stabilizes. Increasing $d$ allows the test to detect finer structural differences between graphs, but the computational cost grows exponentially with $d$. The central question is which dimension allows us to distinguish nonisomorphic graphs. We relate a sufficient such dimension to the structure of a graph $G$ through a rank decomposition over $\mathbb F_2$. The two parameters governing the bound are the maximum cut rank $w$ and the fork load $\ell$, which is the largest sum of the parent and two child cut ranks at a fork in the decomposition tree. We prove that WL in dimension $\max\{\ell+1,2w+2\}$ distinguishes every finite nonempty vertex-coloured graph $G$ with such a decomposition from every graph not isomorphic to $G$. Consequently, for $k\geq1$, graphs of rank width at most $k$ can be distinguished from every nonisomorphic graph in dimension $3k+1$, graphs of linear rank width at most $k$ in dimension $2k+2$, and edgeless graphs in dimension one. Uncoloured Cai-Fürer-Immerman (CFI) constructions give linear lower bounds for both width parameters, so the upper bounds are tight up to constant factors.

Abstract: A graph is $H$-free if it has no induced subgraph isomorphic to $H$. In 2020, Chudnovsky, Goedgebeur, Schaudt, and Zhong characterized all graphs $H$ such that there are only finitely many minimal obstructions to $3$-colouring $H$-free graphs. In general, the minimal obstructions to $k$-colouring $H$-free graphs are the $(k+1)$-vertex-critical $H$-free graphs, those are, the $H$-free graphs $G$ with $χ(G)=k+1$ but $χ(G-v)=k$ for every vertex in $G$. In this paper we complete the characterization for all $k > 4$ by showing that there are onky finitely $k$-vertex-critical $H$-free graphs if and only if $H$ is an induced subgraph of $P_4+\ell P_1$ for some $\ell \geq 0$.

Abstract: We study which graph classes determine every graph with simple spectrum up to isomorphism by homomorphism counts from their members. We show that such a class has unbounded treewidth and Euler genus and contains a graph with the complete graph on seven vertices as a minor. To prove these conditions, we construct pairs of non-isomorphic cospectral graphs with simple spectrum, each bipartite, nonsingular and of maximum degree at most six. The construction attaches rooted trees to Cai-Furer-Immerman graphs and preserves homomorphism indistinguishability over every minor-closed class. We then prove that subcubic homomorphism counts determine every graph whose adjacency matrix has kernel dimension at most one. This extends the known result for nonsingular graphs and applies to all graphs with simple spectrum. We also prove that two graphs have the same subcubic homomorphism counts if and only if they have the same number of vertices and there is a bijective linear isometry between the images of their adjacency matrices that intertwines the adjacency operators and preserves the coordinatewise trilinear forms. Finally, we prove that homomorphism counts from the single-vertex graph and from subdivisions of any sequence of connected cubic graphs have the same distinguishing power as subcubic homomorphism counts if every complete graph is a minor of some member of the sequence.

Abstract: Two graphs $G$ and $H$ are homomorphism indistinguishable over a graph class $\mathcal{F}$ if, for every graph $F \in \mathcal{F}$, the number of homomorphisms from $F$ to $G$ is equal to the number of homomorphisms from $F$ to $H$. Lovász (Acta Mathematica Academiae Scientiarum Hungarica, 1967) showed that two graphs are isomorphic if, and only if, they are homomorphism indistinguishable over all graphs. Subsequently, homomorphism indistinguishability relations of a long list of natural graph classes have been equated with natural graph isomorphism relaxations. Given the wealth of such results, Atserias, Kolaitis, & Wu (LICS 2021) asked for an axiomatic characterisation of homomorphism indistinguishability relations. By exhibiting topological and geometric structure associated with homomorphism indistinguishability, we derive such an axiomatic characterisation. Here, a central ingredient is a novel characterisation of graph parameters of the form $\hom(F, \star)$ for some graph $F$ alternative to a previous result of Lovász & Schrijver (JCTA 2010). Moreover, we investigate the topology of homomorphism indistinguishability and discuss repercussions for the Ulam--Kelly Reconstruction Conjecture.

Abstract: A clique transversal of a graph is a set of vertices intersecting every maximal clique. We prove that deciding whether a graph has an inclusion-wise minimal clique transversal of size at least $k$ is W[1]-hard when parameterized by $k$.

Abstract: For a graph $H$, its $k$-colour vertex Ramsey class is the set of all graphs $G$ such that any colouring of the vertices of $G$ in $k$ colours results in a monochromatic (induced) copy of $H$. We prove that for any $k$, Ramsey classes of any non-isomorphic graphs are distinct.

Abstract: A strong odd $k$-coloring of a graph $G$ is a proper $k$-coloring such that every color appearing in the neighborhood of a non-isolated vertex appears an odd number of times. The minimum $k$ for which $G$ admits a strong odd $k$-coloring is the \emph{strong odd chromatic number}, denoted by $χ_{\text{so}}(G)$, of $G$. Given a graph $G$ and an integer $k$, \textsc{strong odd $k$-colorability} problem asks whether $G$ admits a strong odd $k$-coloring. It is known that STRONG ODD $k$-COLORABILITY is NP-complete in general graphs. In this paper, we prove that the problem is NP-complete on perfect elimination bipartite graphs for $k\geq3$, which is a subclass of bipartite graphs. Furthermore, we show that $χ_{\text{so}}(G)$ is inapproximable within a factor of $O(n^{\frac{1}{2}-\varepsilon})$ for every $\varepsilon>0$. On the positive side, we obtain a linear time algorithm to compute an optimal strong odd coloring for block graphs. From a parameterized perspective, we present an FPT algorithm for STRONG ODD $k$-COLORABILITY when parameterized by treewidth. Moreover, we show that the problem cannot be solved in time $(k-\varepsilon)^{\texttt{tw}}n^{O(1)}$ for every $k\geq3$ and $\varepsilon>0$ when parameterized by treewidth under SETH. Furthermore, we show that STRONG ODD $k$-COLORABILITY does not admit a polynomial kernel when parameterized by feedback vertex set. Lastly, we prove that STRONG ODD $k$-COLORABILITY is W[1]-hard when parameterized by clique-width.

Abstract: We describe one explicit simple graph G on 32 vertices whose adaptive, normal, linear, entropic, and submodular widths are pairwise distinct. We compute all these widths exactly, except for the entropic width, where we only give a lower and upper bound. We use Ingleton's inequality and the Zhang-Yeung inequality for upper bounds, and give explicit constructions of modular, normal, linear, entropic, and non-entropic polymatroids for lower bounds.

Abstract: In this paper, we investigate three fundamental convexity parameters of graphs under cycle convexity, namely the Helly number, Radon number, and rank. We first study the computational complexity of these parameters. For each of these parameters, we consider the associated threshold decision problem of determining whether the parameter of a given graph is at least a prescribed integer. We establish that all three problems are $\NP$-hard and $\W[1]$-hard when parameterized by the threshold. Moreover, we strengthen these results by showing that the $\NP$-hardness persists even when the input is restricted to planar graphs of maximum degree at most $6$. We also focus on the structural properties of connected graphs corresponding to extremal values of these parameters. In particular, we characterize the graph classes for which the three parameters attain the values $n-1$ and $n-2$, where $n$ is the order of $G$.

Abstract: A class of graphs is monadically dependent if one cannot encode all graphs in coloured graphs from the class using a fixed first-order formula, and monadically stable if one cannot even encode arbitrarily long linear orders. Bonnet et al. (ICALP 2025) characterised monadic dependence by flip-separability: for every vertex weighting, boundedly many flips - complementations of the adjacency relation within a vertex subset - make every ball of radius $r$ carry at most an $\varepsilon$-fraction of the weight, so that every set carrying an $\varepsilon$-fraction has two elements pulled apart. We introduce flip-packability: boundedly many graphs, each obtained from the input by boundedly many flips and all determined by the weighting before any set is presented, such that every set carrying an $\varepsilon$-fraction of the weight has $m$ elements pairwise far apart in one of them. The number of flips producing each graph depends on the radius alone; only the number of graphs depends on $\varepsilon$ and $m$. We prove that a class of graphs is flip-packable if and only if it is monadically stable, and $2$-flip-packable, that is, flip-packable with $m=2$, if and only if it is monadically dependent. The passage from two scattered elements to $m$ is thus exactly what separates the two notions. For monadically stable classes we show that the flipped graphs can be computed from the weighting in cubic time. Varying the three parameters of the definition - the sparsifying operation, the radius, and the number $m$ of elements scattered - produces eight known characterisations of sparse and dense graph classes from the same template. In each case $m$ separates a depth-like notion from its width-like relaxation: treedepth from treewidth, shrubdepth from cliquewidth, and monadic stability from monadic dependence.

Abstract: For distinct integers $a$ and $b$, an $\{a,b\}$-edge-weighting assigns $a$ or $b$ to each edge and labels each vertex by the sum of its incident weights. Such a weighting is proper if adjacent vertices receive distinct labels. We prove that, for every fixed pair of distinct integers, deciding whether a proper weighting exists is NP-complete even for simple cubic planar graphs. On planar multigraphs with $m$ edges, we give an exact $2^{O(\sqrt m)}$-time algorithm and, assuming the Exponential Time Hypothesis (ETH), exclude $2^{o(\sqrt m)}$-time algorithms even for simple cubic planar graphs. As a consequence, locally irregular $2$-edge-coloring is NP-complete on simple cubic planar graphs, admits a deterministic $2^{O(\sqrt n)}$-time algorithm on $n$-vertex graphs in this class, and admits no $2^{o(\sqrt n)}$-time algorithm under ETH. For maximizing the number of edges joining vertices with distinct labels, we give a deterministic efficient polynomial-time approximation scheme (EPTAS) on planar multigraphs, a polynomial-time $1/2$-approximation on multigraphs, and APX-completeness even on simple cubic graphs. Extending a partial $\{a,b\}$-edge-weighting to a proper one is NP-complete for every fixed pair even on simple cubic planar bipartite graphs, while it is polynomial-time solvable on trees. The hardness persists even when the prescribed edges form disjoint paths of length $6$ and all edges of each path have the same prescribed weight.

Abstract: We study square-complementary graphs $G$ (satisfying $G^2 \cong \overline{G}$). We show that in such graphs, no two vertices have comparable closed neighborhoods. This implies that nontrivial square-complementary graphs have no simplicial vertices and are not chordal, thus solving two open problems posed in Discrete Mathematics 327 (2014) 62-75. We also show that no nontrivial square-complementary graph is distance-hereditary.

Abstract: The lettericity of a graph $G$ is the smallest size of a set $Σ$ such that there exist $w_1, \ldots, w_{|V(G)|} \in Σ$ and a decoder $D \subseteq Σ^2$ for which $G$ is isomorphic to the letter graph $(\{1, \ldots, |V(G)|\}, \{ij : 1 \le i < j \le |V(G)|, w_iw_j \in D\})$. It took around two decades of the study of lettericity for, in the simpler case of paths, a closed-form expression for its lettericity to be derived; this suggests that the question of whether the lettericity of an arbitrary graph can be computed in polynomial time is nontrivial. Indeed, this question has been raised repeatedly as an open problem in recent literature. We solve this problem by showing that the lettericity problem on arbitrary graphs is \textsf{NP}-complete (Theorem~10). We also prove that the coloring extension problem --- the same problem as lettericity, with the added condition that if $f$ is the isomorphism mapping from $G$ to the letter graph, $w_{f(v)} = χ(v)$ for a given coloring $χ$ of $G$ --- is \textsf{NP}-complete (Theorem~12). We also resolve the open problem of classifying the complexity of the word extension problem, which is the same problem as lettericity except that the $w_i$ are fixed; we show it to be \textsf{NP}-complete (Theorem~13), which, in tandem with our \textsf{NP}-completeness result for coloring extension, contrasts with the known result that when the constraint of the coloring extension problem and the constraint of the word extension problem are both applied to lettericity, lettericity can be decided in polynomial time. Additionally, we use the reduction in the \textsf{NP}-completeness proof to show that unless the Exponential Time Hypothesis is false, there cannot exist a deterministic algorithm to decide whether the lettericity of an $n$-vertex graph is at most~$k$ in time $2^{o(n)}$, even when $n = 6k$ (Theorem~11).

Abstract: A majority coloring (also called an unfriendly partition) of a graph $G$ is a vertex coloring of $G$ in which no vertex has more than half of its neighbors colored with its own color. The least number of colors required for a majority coloring of $G$ is the majority chromatic number $μ(G)$. The majority coloring game, introduced by Bosek--Grytczuk--Jakóbczak (2019), is a two-player Maker--Breaker-type game where the players alternately color vertices while maintaining the majority condition at each vertex. The least number of colors required for the first player to have a winning strategy on $G$ is the majority game chromatic number $μ_g(G)$. In contrast with the static case, Bosek et al. show that $μ_g(G)$ is unbounded in general, while $μ_g(G) \le \mathrm{col}_g(G)$, where $\mathrm{col}_g(G)$ is the game coloring number of $G$. It is known that for any acyclic graph $G$, $\mathrm{col}_g(G) \le 4$, and hence $μ_g(G) \le 4$. We improve this bound by showing that $μ_g(G) \le 3$ for any acyclic graph $G$ of maximum degree at most $4$. We also show that $μ_g(G) \le 2$ if $G$ is a path, a star, or a complete graph, improving results of Bosek et al. We also initiate the study of the computational complexity of the majority coloring game. We show that the pre-coloring extension problem for majority coloring on $G$ with a palette of $χ(G)$ colors is NP-complete, and that its game version is PSPACE-complete. Furthermore, the problem remains NP-complete, and its game version remains PSPACE-complete, even with a palette of $2$ colors.

Abstract: The Pfaffian number of a graph is the minimum number of Pfaffians needed to obtain its perfect-matching polynomial by linear combination. In 2009, Norine conjectured that every Pfaffian number is a power of four. Miranda and Lucchesi disproved this conjecture in 2011 by constructing a graph of Pfaffian number six, and conjectured instead that every nontrivial Pfaffian number is even. We disprove their conjecture by proving that the Pfaffian number of $K_{3,3}\sqcup K_{3,3}$ is 13. We also construct a connected cubic bipartite matching-covered graph with Pfaffian number 13.

Abstract: The notion of induced packing treewidth aims to unify classes defined by forbidden induced subgraphs or induced minors with classes defined by the existence of certain structured tree decompositions. For a graph $H$, \emph{induced $H$-packing treewidth}, denoted by $\treepi_{H}$, is a tree-decomposition-based graph parameter that, for each bag, measures the maximum number of pairwise anticomplete induced copies of $H$ intersecting that bag. This notion generalizes some previously studied parameters: when $H=P_1$, it is equivalent to tree-independence number, and when $H=P_2$, it is equivalent to induced matching treewidth. We prove the following: \begin{itemize}[itemsep=2mm,leftmargin=6mm] \item For all $a,t\in \mathbb{N}$, $K_{a,a}$-free graphs of bounded induced $P_t$-packing treewidth have bounded tree-independence number. This extends the previous result of Abrishami et al. [SIAM J. Discrete Math., 2025] for $t=2$, and a result of Hajebi and Spirkl who showed that $(P_t,K_{a,a})$-free graphs have bounded tree-independence number. \item If $H$ is any fixed path or a star, then the class of graphs of bounded induced $H$-packing treewidth is $χ$-bounded. Again, this extends the previous result of Abrishami et al. [SIAM J. Discrete Math., 2025] for $H=P_2$. \item Finally, we study the relationship between induced packing treewidth and \emph{sim-width}, a width parameter based on branch decompositions. We show that, although \emph{sim-width} and induced $P_3$-packing treewidth are incomparable, graphs of bounded sim-width that exclude all \emph{$H$-obstructions}---certain graphs that force large induced $H$-packing treewidth---have bounded induced $H$-packing treewidth. This simultaneously generalizes and resolves questions posed by Abrishami et al. [SIAM J. Discrete Math., 2025] and Brettell et al. [European J. Comb., 2025]. \end{itemize}

Abstract: We prove that counting perfect matchings is $\#P$-complete under polynomial-time Turing reductions on each of three classes of simple, unweighted graphs: monotone graphs, unit interval graphs, and chordal permutation graphs. The monotone result settles the exact-counting complexity left open by Dyer, Jerrum, and Müller (JACM 2017), complementing their rapid-mixing theorem. Inspired by quantum circuits, our reductions implement a circuit simulation using globally coupled matching-transfer operators. The key construction is an exact projection, implemented by a polynomial-length sequence of normalized transfers, that restores tensor-product locality and makes encoded gates composable. Interpolation-based cancellation then reduces circuit evaluation to unweighted perfect-matching counts in all three classes. We also place Dyer and Müller's class QChains within the distance-hereditary graphs and give an $O(n^2)$-arithmetic-operation counting algorithm for the latter, improving the $O(n^4)$ bound obtainable from Curticapean and Marx (SODA 2016). Together with prior results, these advances complete the exact-counting classification of the graph classes in Dyer and Müller's diagram (SIDMA 2019).

Abstract: The forced colouring function of a graph gives the probability that a random assignment of colours to a random subset of vertices can be extended, by a simple local process called forcing, to give a proper colouring of the whole graph using the same set of available colours. This is a polynomial for each fixed number of colours, and was introduced as a subject for research on the general theory of graph polynomials. In this paper we establish its fundamental properties and give combinatorial interpretations of its derivatives at two particular points. We also prove that the problem of computing its value at any specific point in a certain interval is #P-hard.

Abstract: A partial matrix has entries in $\{0,1,\star\}$, and a disambiguation replaces each $\star$ by $0$ or $1$. We construct partial matrices whose fully specified submatrices satisfy strong restrictions, yet every disambiguation contains every binary matrix of a prescribed size. Our first result answers a question of Alon, Hanneke, Holzman and Moran on the disambiguation of linear classifiers with margin. For $0<\varepsilon<π/2$, let $M_\varepsilon^d$ be the partial matrix indexed by points of the unit sphere $\mathbb S^d$, with entry $0$ for pairs at spherical distance at most $\varepsilon$, $1$ for pairs at distance at least $π-\varepsilon$, and $\star$ otherwise. Although these matrices have VC-dimension bounded independently of $d$, we prove that every disambiguation contains every binary $k\times k$ matrix once $d$ is sufficiently large. This also yields a partial concept class of Littlestone dimension $1$ with no disambiguation of finite VC-dimension. We also construct, for every $k$, a finite partial matrix whose fully specified $2\times2$ submatrices are all constant, while every disambiguation contains every binary $k\times k$ matrix. A symmetric analogue holds for partial graphs: for every $k$, there exists a partial graph of VC-dimension at most $1$ whose fully specified induced subgraphs are all cliques or stable sets, yet every disambiguation contains every $k$-vertex graph as an induced subgraph. A disambiguation can be viewed as a $2$-coloring of the unspecified entries, making Ramsey theory a natural framework for forcing prescribed patterns. Our proofs draw on two recent Ramsey theorems: the geometric argument uses Pálvölgyi's Dense Block theorem, while the combinatorial constructions rely on the girth Ramsey theorem of Reiher and Rödl, a suitable strengthening of the induced Ramsey theorem.

Abstract: In this work, we consider the problem of maintaining an approximate value of degeneracy of a given dynamic $n$-vertex graph $G$ updated by edge insertions and deletions. From the work of Christiansen and Rotenberg [ICALP 2022], it follows that one can design a dynamic data structure for this problem with worst-case update time $\text{poly}(d_{\mathrm{max}}, \log n)$ that maintains an integer between $d$ and $2d+3$ where $d$ is the degeneracy of $G$, under the assumption that $d$ never exceeds $d_{\mathrm{max}}$. We complement their result by providing a conditional lower bound: we prove that, unless SETH fails, for any $\varepsilon, δ> 0$, $k \in \mathbb{N}$, and function $f\colon \mathbb{N}\to \mathbb{N}$, there is no data structure which maintains a $(2-\varepsilon)$-approximation of the degeneracy of $G$ with initialization time $f(d_{\mathrm{max}})\cdot n^k$ and amortized update time $f(d_{\mathrm{max}})\cdot n^{1-δ}$.

Abstract: Network motifs and colored local interaction patterns provide a useful way to describe the structure of complex networks. Motivated by an inverse realization perspective, we study the problem of assigning colors to the vertices of a fixed graph so that its edges realize a prescribed multiset of colored pairwise interactions. This problem is formalized by the Matching-Match Puzzle, introduced by Iburi and Uehara. We investigate how its computational complexity depends on the number of colors and on the structure of the host graph. We first prove that Matching-Match is NP-complete with only two colors, even when no vertex is precolored and the graph has maximum degree three. We then focus on spiders. We show that the problem is W[1]-hard parameterized by the number of colors even on spiders with only the body precolored. In contrast, for spiders whose legs have length at most two, we give a fixed-parameter tractable algorithm parameterized by the number of colors, allowing arbitrary precoloring. Finally, we prove NP-completeness for spiders whose legs all have length exactly three when precoloring is allowed.

Abstract: Coloring Discovery asks whether a possibly improper initial coloring can be made proper within a prescribed number of allowed changes. We study the parameterized complexity of three modification step models that were studied previously in the literature: recoloring one vertex (color flipping), swapping the colors of arbitrary vertices (color swapping), and swapping colors only across an edge (color sliding). For color flipping, we give exact fixed-parameter algorithms for the parameters vertex cover and distance to complete. For color swapping, we obtain fixed-parameter tractability for the parameter vertex cover plus the number of colors. Our lower bounds show W[1]-hardness for treedepth plus feedback vertex set in the color flipping model and for the number of colors plus bandwidth or distance to disjoint paths in the swapping and sliding models. All three variants remain NP-complete with four colors on graphs of diameter two.

Abstract: In this paper, we study \textsc{MaxMMP}, an optimization variant of the Matching-Match Puzzle introduced by Iburi and Uehara (FUN 2024). Given a graph, a partial vertex coloring, and a multiset of colored sticks, the goal is to complete the coloring and assign the sticks to graph edges so as to maximize the number of satisfied edges. We first prove that \textsc{MaxMMP} is APX-hard by an reduction from \textsc{Max-Cut}. The hardness already holds with two colors, no precolored vertices, and only bichromatic sticks. We then give a simple deterministic $\frac{2}{c(c+1)}$-approximation for completely uncolored instances, improving to $\frac{2}{c(c-1)}$ when all sticks are bichromatic. Next, we obtain a randomized $\frac{1-\frac{1}{e}}{2c}$-approximation for arbitrary instances with $c$ colors by reducing the remaining coloring choices to monotone submodular maximization under a partition matroid. On bipartite graphs, the approximation ratio improves to $\frac{1-\frac{1}{e}}{c}$. For every fixed $c$, we further obtain deterministic $\frac{1}{2c}$ and $\frac{1}{c}$-approximations on general and bipartite graphs, respectively, in time $n^{O(c^2)}$. Finally, for every fixed number of colors, we show that \textsc{MaxMMP} can be solved exactly in time $n^{O(c^2)}$ on trees and on cographs.