arXiv subjects

Carla Groenland

Publications and source records attributed to Carla Groenland.

At least 19 recordsLinked to original sources

arXiv2026

Faithful universal graphs for minor-closed classes

It was proved by Huynh, Mohar, Šámal, Thomassen and Wood in 2021 that any countable graph containing every countable planar graph as a subgraph has an infinite clique minor. We prove a finite, quantitative version of this result: for fixed $t$, if a graph $G$ is $K_t$-minor-free and contains every $n$-vertex planar graph as a subgraph, then $G$ has $2^{Ω(n)}$ vertices. On the other hand, we construct a polynomial size $K_4$-minor-free graph containing every $n$-vertex tree as an induced subgraph, and a polynomial size $K_7$-minor-free graph containing every $n$-vertex $K_4$-minor-free graph as induced subgraph. This answers several problems raised recently by Bergold, Iršič, Lauff, Orthaber, Scheucher and Wesolek. We study more generally the order of universal graphs for various classes (of graphs of bounded degree, treedepth, pathwidth, or treewidth), if the universal graphs retain some of the structure of the original class.

math.CO↗

arXiv2026

Faster 3-colouring algorithm for graphs of diameter 3

We show that given an $n$-vertex graph $G$ of diameter 3 we can decide if $G$ is $3$-colourable in time $2^{O(n^{2/3-\varepsilon})}$ for any $\varepsilon < 1/33$. This improves on the previous best algorithm of $2^{O((n\log n)^{2/3})}$ from D\k{e}bski, Piecyk and Rz\k{a}\.zewski [Faster 3-coloring of small-diameter graphs, ESA 2021].

math.CO↗

arXiv2026

Trade-off between spread and width for tree decompositions

The spread of a vertex $v$ in a tree decomposition is the number of bags that contain $v$. We study the trade-off between spread and width in tree decompositions, answering every open question from Wood [arXiv:2509.01140]. First, Wood asked for the infimum of $c > 0$ such that there exists $c'$ such that each graph $G$ has a tree decomposition of width $c \cdot tw(G)$ in which each vertex $v$ has spread at most $c'(d(v)+1)$. We show that the answer is $3$. Second, we prove a conjecture of Wood, stating that every tree-decomposition of the $(n \times n)$-grid with width $n$ has a vertex with spread $\Omega(n)$. Finally, we answer the last question of Wood by showing that near-optimal average spread can be achieved simultaneously with width $O(tw(G))$.

math.CO↗

arXiv2025

Improved exploration of temporal graphs

A temporal graph $G$ is a sequence $(G_t)_{t \in I}$ of graphs on the same vertex set of size $n$. The \emph{temporal exploration problem} asks for the length of the shortest sequence of vertices that starts at a given vertex, visits every vertex, and at each time step $t$ either stays at the current vertex or moves to an adjacent vertex in $G_t$. Bounds on the length of a shortest temporal exploration have been investigated extensively. Perhaps the most fundamental case is when each graph $G_t$ is connected and has bounded maximum degree. In this setting, Erlebach, Kammer, Luo, Sajenko, and Spooner [ICALP 2019] showed that there exists an exploration of $G$ in $\mathcal{O}(n^{7/4})$ time steps. We significantly improve this bound by showing that $\mathcal{O}(n^{3/2} \sqrt{\log n})$ time steps suffice. In fact, we deduce this result from a much more general statement. Let the \emph{average temporal maximum degree} $D$ of $G$ be the average of $\max_{t \in I} d_{G_t}(v)$ over all vertices $v \in V(G)$, where $d_{G_t}(v)$ denotes the degree of $v$ in $G_t$. If each graph $G_t$ is connected, we show that there exists an exploration of $G$ in $\mathcal{O}(n^{3/2} \sqrt{D \log n})$ time steps. In particular, this gives the first subquadratic upper bound when the underlying graph has bounded average degree. As a special case, this also improves the previous best bounds when the underlying graph is planar or has bounded treewidth and provides a unified approach for all of these settings. Our bound is subquadratic already when $D=o(n/\log n)$.

cs.DS↗

arXiv2025

Longest cycles in vertex-transitive and highly connected graphs

We present progress on three old conjectures about longest paths and cycles in graphs. The first pair of conjectures, due to Lovász from 1969 and Thomassen from 1978, respectively, states that all connected vertex-transitive graphs contain a Hamiltonian path, and that all sufficiently large such graphs even contain a Hamiltonian cycle. The third conjecture, due to Smith from 1984, states that for $r\ge 2$ in every $r$-connected graph any two longest cycles intersect in at least $r$ vertices. In this paper, we prove a new lemma about the intersection of longest cycles in a graph which can be used to improve the best known bounds towards all the aforementioned conjectures: First, we show that every connected vertex-transitive graph on $n\geq 3$ vertices contains a cycle (and hence path) of length at least $Ω(n^{13/21})$, improving on $Ω(n^{3/5})$ from [DeVos, \emph{arXiv:2302:04255}, 2023]. Second, we show that in every $r$-connected graph with $r\geq 2$, any two longest cycles meet in at least $Ω(r^{5/8})$ vertices, improving on $Ω(r^{3/5})$ from [Chen, Faudree and Gould, \emph{J. Combin. Theory, Ser.~ B}, 1998]. Our proof combines combinatorial arguments, computer-search and linear programming.

math.CO↗

arXiv2025

Short reachability networks

We investigate the following generalisation of permutation networks. We say a sequence $T=(T_1,\dots,T_\ell)$ of transpositions in $S_n$ forms a $t$-reachability network if, for every choice of $t$ distinct points $x_1, \dots, x_t\in \{1,\dots,n\}$, there is a subsequence of $T$ whose composition maps $j$ to $x_j$ for every $1\leq j\leq t$. When $t=n$, any permutation in $S_n$ can be created and $T$ is a permutation network. Waksman [JACM, 1968] showed that the shortest permutation networks have length about $n \log_2(n)$. In this paper, we investigate the shortest $t$-reachability networks for other values of $t$. Our main result settles the case of $t=2$: the shortest $2$-reachability network has length $\lceil 3n/2\rceil-2 $. For fixed $t \geq 3$, we give a simple randomised construction which shows that there exist $t$-reachability networks with $(2+o_t(1))n$ transpositions. We also study the effect of restricting to star-transpositions, i.e. restricting all transpositions to have the form $(1, \cdot)$.

math.CO↗

arXiv2025

Circular sorting, strong complete mappings and wreath product constructions

We continue the study of Adin, Alon and Roichman [arXiv:2502.14398, 2025] on the number of steps required to sort $n$ labelled points on a circle by transpositions. Imagine that the vertices of a cycle of length $n$ are labelled by the elements $1,\dots,n$. We are allowed to change this labelling by swapping the labels of any two vertices on the cycle. How many swaps are needed to obtain a labelling that has the elements $1,\dots,n$ in clockwise order? We provide evidence for their conjecture that at most $n-3$ transpositions are needed to sort a circular permutation when $n$ is not prime. We prove this conjecture when $2\mid n$ or $3\mid n$ and when restricting to permutations given by a polynomial over $\mathbb{Z}_n$. We also provide various algebraic constructions of circular permutations that take many transpositions to sort, most notably providing one that matches our upper bound when $n=3p$ for $p$ an odd prime, and disproving their second conjecture by providing non-affine circular permutations that require $n-2$ transpositions (for $n$ prime). We also improve the lower bounds for some sequences of composite numbers. Finally, we improve the bounds for small $n$ computationally. In particular, we prove a tight upper bound for $n=25$ via an exhaustive computer search using a new connection between this problem and strong complete mappings.

math.CO↗

arXiv2025

Smaller universal posets

We show that there is a constant $C>0$ such that for each integer $n\geq 1$, there is a poset on at most $2^{2n/3+C\sqrt{n}}$ elements that contains each $n$-element poset as an (induced) subposet.

math.CO↗

arXiv2025

Infinite induced-saturated graphs

A graph $G$ is $H$-induced-saturated if $G$ is $H$-free but deleting any edge or adding any edge creates an induced copy of $H$. There are non-trivial graphs $H$, such as $P_4$, for which no finite $H$-induced-saturated graph $G$ exists. We show that for every finite graph $H$ that is not a clique or an independent set, there always exists a countable $H$-induced-saturated graph. In fact, we show that a far stronger property can be achieved: there is a countably infinite $H$-free graph $G$ such that any graph $G'\ne G$ obtained by making a locally finite set of changes to $G$ contains a copy of $H$.

math.CO↗

arXiv2025

Note on extremal problems about connected subgraph sums

For a graph $G$ with vertex assignment $c:V(G)\to \mathbb{Z}^+$, we define $\sum_{v\in V(H)}c(v)$ for $H$ a connected subgraph of $G$ as a connected subgraph sum of $G$. We study the set $S(G,c)$ of connected subgraph sums and, in particular, resolve a problem posed by Solomon Lo in a strong form. We show that for each $n$-vertex graph, there is a vertex assignment $c:V(G)\to \{1,\dots,12n^2\}$ such that for every $n$-vertex graph $G'\not\cong G$ and vertex assignment $c'$ for $G'$, the corresponding collections of connected subgraph sums are different (i.e., $S(G,c)\neq S(G',c')$). We also provide some remarks on vertex assignments of a graph $G$ for which all connected subgraph sums are different.

math.CO↗

arXiv2025

On the parameterized complexity of computing tree-partitions

We study the parameterized complexity of computing the tree-partition-width, a graph parameter equivalent to treewidth on graphs of bounded maximum degree. On one hand, we can obtain approximations of the tree-partition-width efficiently: we show that there is an algorithm that, given an $n$-vertex graph $G$ and an integer $k$, constructs a tree-partition of width $O(k^7)$ for $G$ or reports that $G$ has tree-partition-width more than $k$, in time $k^{O(1)}n^2$. We can improve slightly on the approximation factor by sacrificing the dependence on $k$, or on $n$. On the other hand, we show the problem of computing tree-partition-width exactly is XALP-complete, which implies that it is $W[t]$-hard for all $t$. We deduce XALP-completeness of the problem of computing the domino treewidth. Next, we adapt some known results on the parameter tree-partition-width and the topological minor relation, and use them to compare tree-partition-width to tree-cut width. Finally, for the related parameter weighted tree-partition-width, we give a similar approximation algorithm (with ratio now $O(k^{15})$) and show XALP-completeness for the special case where vertices and edges have weight 1.

cs.DM↗

arXiv2024

Optimal distance query reconstruction for graphs without long induced cycles

Given access to the vertex set $V$ of a connected graph $G=(V,E)$ and an oracle that given two vertices $u,v\in V$, returns the shortest path distance between $u$ and $v$, how many queries are needed to reconstruct $E$? Firstly, we show that randomised algorithms need to use at least $\frac1{200} Δn\log_Δn$ queries in expectation in order to reconstruct $n$-vertex trees of maximum degree $Δ$. The best previous lower bound (for graphs of bounded maximum degree) was an information-theoretic lower bound of $Ω(n\log n/\log \log n)$. Our randomised lower bound is also the first to break through the information-theoretic barrier for related query models including distance queries for phylogenetic trees, membership queries for learning partitions and path queries in directed trees. Secondly, we provide a simple deterministic algorithm to reconstruct trees using $Δn\log_Δn+(Δ+2)n$ distance queries. This proves that our lower bound is optimal up to a multiplicative constant. We extend our algorithm to reconstruct graphs without induced cycles of length at least $k$ using $O_{Δ,k}(n\log n)$ queries. Our lower bound is therefore tight for a wide range of tree-like graphs, such as chordal graphs, permutation graphs and AT-free graphs. The previously best randomised algorithm for chordal graphs used $O_Δ(n\log^2 n)$ queries in expectation, so we improve by a $(\log n)$-factor for this graph class.

cs.DS↗

arXiv2024

Quasi-linear distance query reconstruction for graphs of bounded treelength

In distance query reconstruction, we wish to reconstruct the edge set of a hidden graph by asking as few distance queries as possible to an oracle. Given two vertices $u$ and $v$, the oracle returns the shortest path distance between $u$ and $v$ in the graph. The length of a tree decomposition is the maximum distance between two vertices contained in the same bag. The treelength of a graph is defined as the minimum length of a tree decomposition of this graph. We present an algorithm to reconstruct an $n$-vertex connected graph $G$ parameterized by maximum degree $Δ$ and treelength $k$ in $O_{k,Δ}(n \log^2 n)$ queries (in expectation). This is the first algorithm to achieve quasi-linear complexity for this class of graphs. The proof goes through a new lemma that could give independent insight on graphs of bounded treelength.

cs.DS↗

arXiv2024

A Polynomial Time Algorithm for Steiner Tree when Terminals Avoid a $K_4$-Minor

We study a special case of the Steiner Tree problem in which the input graph does not have a minor model of a complete graph on 4 vertices for which all branch sets contain a terminal. We show that this problem can be solved in $O(n^4)$ time, where $n$ denotes the number of vertices in the input graph. This generalizes a seminal paper by Erickson et al. [Math. Oper. Res., 1987] that solves Steiner tree on planar graphs with all terminals on one face in polynomial time.

cs.DS↗

arXiv2024

Counting graphic sequences via integrated random walks

Given an integer $n$, let $G(n)$ be the number of integer sequences $n-1\ge d_1\ge d_2\ge\dotsb\ge d_n\ge 0$ that are the degree sequence of some graph. We show that $G(n)=(c+o(1))4^n/n^{3/4}$ for some constant $c>0$, improving both the previously best upper and lower bounds by a factor of $n^{1/4}$ (up to polylog-factors). Additionally, we answer a question of Royle, extend the values of $n$ for which the exact value of $G(n)$ is known from $n\le290$ to $n\le 1651$ and determine the asymptotic probability that the integral of a (lazy) simple symmetric random walk bridge remains non-negative.

math.CO↗

arXiv2024

A Polynomial Upper Bound for Poset Saturation

Given a finite poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced copy of $\mathcal P$. The induced saturation number of $\mathcal P$, denoted by $\text{sat}^*(n,\mathcal P)$, is the size of the smallest $\mathcal P$-saturated family with ground set $[n]$. In this paper we prove that the saturation number for any given poset grows at worst polynomially. More precisely, we show that $\text{sat}^*(n, \mathcal P)=O(n^c)$, where $c\leq|\mathcal{P}|^2/4+1$ is a constant depending on $\mathcal P$ only. We obtain this result by bounding the VC-dimension of our family.

math.CO↗

arXiv2024

On the Complexity of Problems on Tree-structured Graphs

In this paper, we introduce a new class of parameterized problems, which we call XALP: the class of all parameterized problems that can be solved in $f(k)n^{O(1)}$ time and $f(k)\log n$ space on a non-deterministic Turing Machine with access to an auxiliary stack (with only top element lookup allowed). Various natural problems on `tree-structured graphs' are complete for this class: we show that List Colouring and All-or-Nothing Flow parameterized by treewidth are XALP-complete. Moreover, Independent Set and Dominating Set parameterized by treewidth divided by $\log n$, and Max Cut parameterized by cliquewidth are also XALP-complete. Besides finding a `natural home' for these problems, we also pave the road for future reductions. We give a number of equivalent characterisations of the class XALP, e.g., XALP is the class of problems solvable by an Alternating Turing Machine whose runs have tree size at most $f(k)n^{O(1)}$ and use $f(k)\log n$ space. Moreover, we introduce `tree-shaped' variants of Weighted CNF-Satisfiability and Multicolour Clique that are XALP-complete.

cs.CC↗

arXiv2023

Towards Tight Bounds for the Graph Homomorphism Problem Parameterized by Cutwidth via Asymptotic Rank Parameters

A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. In the graph homomorphism problem, denoted by $Hom(H)$, the graph $H$ is fixed and we need to determine if there exists a homomorphism from an instance graph $G$ to $H$. We study the complexity of the problem parameterized by the cutwidth of $G$. We aim, for each $H$, for algorithms for $Hom(H)$ running in time $c_H^k n^{\mathcal{O}(1)}$ and matching lower bounds that exclude $c_H^{k \cdot o(1)}n^{\mathcal{O}(1)}$ or $c_H^{k(1-Ω(1))}n^{\mathcal{O}(1)}$ time algorithms under the (Strong) Exponential Time Hypothesis. In the paper we introduce a new parameter that we call $\mathrm{mimsup}(H)$. Our main contribution is strong evidence of a close connection between $c_H$ and $\mathrm{mimsup}(H)$: * an information-theoretic argument that the number of states needed in a natural dynamic programming algorithm is at most $\mathrm{mimsup}(H)^k$, * lower bounds that show that for almost all graphs $H$ indeed we have $c_H \geq \mathrm{mimsup}(H)$, assuming the (Strong) Exponential-Time Hypothesis, and * an algorithm with running time $\exp ( {\mathcal{O}( \mathrm{mimsup}(H) \cdot k \log k)}) n^{\mathcal{O}(1)}$. The parameter $\mathrm{mimsup}(H)$ can be thought of as the $p$-th root of the maximum induced matching number in the graph obtained by multiplying $p$ copies of $H$ via certain graph product, where $p$ tends to infinity. It can also be defined as an asymptotic rank parameter of the adjacency matrix of $H$. Our results tightly link the parameterized complexity of a problem to such an asymptotic rank parameter for the first time.

cs.DM↗