arXiv subjects

Rose McCarty

Publications and source records attributed to Rose McCarty.

At least 19 recordsLinked to original sources

arXiv2026

Neighborhood Complexity and Radius-1 Merge-Width in Monadically Dependent Graph Classes

Monadic dependence is a proposed structural dividing line for fixed-parameter tractability of first-order model checking on hereditary graph classes. A graph class is \emph{monadically dependent} if the class of all graphs cannot be interpreted in its vertex-colored members using a fixed first-order formula. We prove two structural consequences of monadic dependence. First, every monadically dependent class has \emph{almost linear neighborhood complexity}: for every graph $G$ in the class and every set $A\subseteq V(G)$, the family $\{N_G(v)\cap A : v\in V(G)\}$ has size $|A|^{1+o(1)}$. Second, every $n$-vertex graph in a monadically dependent class has radius-1 merge-width $n^{o(1)}$. Here, merge-width is the decomposition parameter of Dreier and Toru\'nczyk based on construction sequences; its radius-$r$ version measures local reachability among parts through already resolved pairs. This settles the radius-1 case of the conjectured connection between monadic dependence and almost bounded merge-width and provides the first decomposition-based structural description of monadically dependent graph classes. Our proof is algorithmic: we give an $\mathcal{O}(n^5)$-time algorithm that, given an $n$-vertex graph $G$ such that $|\{N_G(v)\cap A : v\in V(G)\}|\le O(|A|^d)$ for every $A\subseteq V(G)$, computes a construction sequence witnessing radius-1 merge-width $\mathcal{O}(n^{1-1/d}\log n)$.

cs.DM↗

arXiv2026

A combinatorial framework for clustering graph states: Algorithms and hardness for rank-integrity

We introduce a new notion of distance between two graph states $|G\rangle$ and $|G'\rangle$ on the same set of qubits. This distance is the minimum number of ancilla qubits in a graph state $|\widehat{G}\rangle$ from which both $|G\rangle$ and $|G'\rangle$ can be ``easily prepared''. (When preparing graph states, we are only allowed to use one-qubit Clifford gates, one-qubit Pauli measurements, and classical communication.) We give a graphical description of this distance through the lens of vertex-minors. We then show how this distance yields quantum network analogs of many graph edit-distance problems. Using this framework, we develop classical algorithms for identifying the ``highly entangled clusters'' of a graph state $|G\rangle$. The ancilla integrity problem asks, given a graph $G$ and integer $k$, for the minimum -- over all graph states $|G'\rangle$ with distance at most $k$ from $|G\rangle$ -- of the maximum component size of $G'$. Up to a factor of $2$ in the number of ancilla qubits, this problem is equivalent to rank integrity, where the distance between $G$ and $G'$ is instead the minimum rank of the sum of their adjacency matrices over $\text{GF}(2)$. We prove that rank integrity is XP parameterized by $k$. We also prove the complementary hardness result that rank integrity is W[1]-hard in $k$. Finally, we give an explicit $\mathcal{O}(n^6)$-time algorithm for ancilla integrity when $G$ has $n$ vertices and $k=1$.

cs.DS↗

arXiv2026

Biclique decompositions from Welzl orders

A biclique decomposition of a graph is a partition of its edges into complete bipartite subgraphs. We consider graphs whose vertices can be ordered such that the neighborhood of every vertex is the union of a sublinear number of intervals. We observe that these graphs admit compact representations in the form of biclique decompositions of small size. Here, the size of a decomposition is measured as the sum of the number of vertices of its bicliques. Combining this result with the existence of suitable vertex orderings for graphs of low neighborhood complexity, as proven by Welzl in 1988, we recover and extend several known results up to logarithmic factors. These results include upper bounds on the Zarankiewicz problem, matrix multiplication, quantum circuit complexity, and shortest path algorithms in ``well-structured'' instances.

math.CO↗

arXiv2026

The structure of group-labeled graphs forbidding an immersion

A $\Gamma$-labeled graph is an oriented graph with edges invertibly labeled by a group $\Gamma$. We prove a structure theorem for $\Gamma$-labeled graphs which forbid a fixed $\Gamma$-labeled graph as an immersion, for any finite $\Gamma$. Roughly, we show that such graphs admit a tree-cut decomposition in which every bag either contains few high degree vertices or is nearly signed over a proper subgroup of $\Gamma$.

math.CO↗

arXiv2026

The Structure of Circle Graph States

Circle graph states are a structurally important family of graph states. The family's entanglement is a priori high enough to allow for universal measurement-based quantum computation (MBQC); however, MBQC on circle graph states is actually efficiently classically simulable. In this work, we paint a detailed picture of the local equivalence of circle graph states. First, we consider the class of all graph states that are local unitary (LU)-equivalent to circle graph states. In graph-theoretical terms, this LU-equivalence class is the set of all graphs reachable from the family of circle graphs by applying $r$-local complementations. We prove that the only graph states that are LU-equivalent to circle graph states are circle graph states themselves: circle graphs are closed under $r$-local complementation. Second, we show that bipartite circle graph states, i.e., 2-colorable circle graph states, are in one-to-one correspondence with planar code states, on which MBQC is known to be efficiently classically simulable. Leveraging this correspondence, we present alternative, simple proofs that (1) if a planar code state is LU-equivalent to a stabilizer state, they are in fact local Clifford (LC)-equivalent to it and that (2) MBQC on all circle graph states is efficiently classically simulable. Third and finally, we demonstrate that the problem of counting the number of graph states LU-equivalent to a given graph state is $\#\mathsf{P}$-hard.

quant-ph↗

arXiv2026

Graphs whose Eulerian trails have unique labels

Consider an undirected graph whose edges are labeled invertibly in a group. When does every Eulerian trail from one fixed vertex to another have the same label? We give a precise structural answer to this question. Essentially, we show that each ``$3$-connected part'' is labeled over a group which is isomorphic to $\mathbb{Z}_2^k$ for some $k$. We also show that the algorithmic problem admits a polynomial-time reduction to the word problem for the group.

math.CO↗

arXiv2025

Quantum Graph States: Bridging Classical Theory and Quantum Innovation, Workshop Summary

This workshop brought together experts in classical graph theory and quantum information science to explore the intersection of these fields, with a focus on quantum graph states and their applications in computing, networking, and sensing. The sessions highlighted the foundational role of graph-theoretic structure, such as rank-width, vertex-minors, and hypergraphs, in enabling measurement-based quantum computation, fault-tolerant architectures, and distributed quantum sensing. Key challenges identified include the need for scalable entanglement generation, robust benchmarking methods, and deeper theoretical understanding of generalized graph states. The workshop concluded with targeted research recommendations, emphasizing interdisciplinary collaboration to address open problems in entanglement structure, simulation complexity, and experimental realization across diverse quantum platforms.

quant-ph↗

arXiv2025

Odd coloring graphs with linear neighborhood complexity

We prove that any class of graphs with linear neighborhood complexity has bounded improper odd chromatic number. As a result, if $\mathcal{G}$ is the class of all circle graphs, or if $\mathcal{G}$ is any class with bounded twin-width, bounded merge-width, or a forbidden vertex-minor, then $\mathcal{G}$ is $\chi_{\mathrm{o}}$-bounded.

math.CO↗

arXiv2025

The Erd\H{o}s-P\'{o}sa property for circle graphs as vertex-minors

We prove that for any circle graph $H$ with at least one edge and for any positive integer $k$, there exists an integer $t=t(k,H)$ so that every graph $G$ either has a vertex-minor isomorphic to the disjoint union of $k$ copies of $H$, or has a $t$-perturbation with no vertex-minor isomorphic to $H$. Using the same techniques, we also prove that for any planar multigraph $H$, every binary matroid either has a minor isomorphic to the cycle matroid of $kH$, or is a low-rank perturbation of a binary matroid with no minor isomorphic to the cycle matroid of $H$.

math.CO↗

arXiv2025

Girth in $GF(q)$-representable matroids

We prove a conjecture of Geelen, Gerards, and Whittle that for any finite field $GF(q)$ and any integer $t$, every cosimple $GF(q)$-representable matroid with sufficiently large girth contains either $M(K_t)$ or $M(K_t)^*$ as a minor.

math.CO↗

arXiv2025

Strongly sublinear separators and bounded asymptotic dimension for sphere intersection graphs

In this paper, we consider the class $\mathcal{C}^d$ of sphere intersection graphs in $\mathbb{R}^d$ for $d \geq 2$. We show that for each integer $t$, the class of all graphs in $\mathcal{C}^d$ that exclude $K_{t,t}$ as a subgraph has strongly sublinear separators. We also prove that $\mathcal{C}^d$ has asymptotic dimension at most $2d+2$.

math.CO↗

arXiv2025

The Structural Complexity of Matrix-Vector Multiplication

We consider the problem of preprocessing an $n\times n$ matrix $\mathbf{M}$, and supporting queries that, for any vector $v$, returns the matrix-vector product $\mathbf{M} v$. This problem has been extensively studied in both theory and practice: on one side, practitioners have developed algorithms that are highly efficient in practice, whereas on the other side, theoreticians have proven that the problem cannot be solved faster than naive multiplication in the worst-case. This lower bound holds even in the average-case, implying that existing average-case analyses cannot explain this gap between theory and practice. Hence, we study the problem for \emph{structured} matrices. We show that for $n\times n$ Boolean matrices of VC-dimension $d$, the matrix-vector multiplication problem can be solved with $\widetilde{O}(n^2)$ preprocessing and $\widetilde{O}(n^{2-1/d})$ query time. Given the low constant VC-dimensions observed in most real-world data, our results posit an explanation for why the problem can be solved so much faster in practice. Furthermore, we show how to extend this result to the non-Boolean setting with the Pollard pseudodimension. Our results yield the first non-trivial upper bounds for many applications. In previous works, the online matrix-vector (OMv) hypothesis (conjecturing that quadratic time is needed per query, even over the boolean semi-ring) was used to prove many conditional lower bounds, showing that it is impossible to compute and maintain high-accuracy estimates for effective resistance, Laplacian solvers, shortest paths, and triangle detection in graphs subject to node insertions and deletions in subquadratic time. Yet, via a reduction to our matrix-vector multiplication result, we show we can maintain these problems efficiently if the input is structured, providing the first subquadratic upper bounds in the high-accuracy regime.

cs.DS↗

arXiv2024

Fat minors cannot be thinned (by quasi-isometries)

We disprove the conjecture of Georgakopoulos and Papasoglu that a length space (or graph) with no $K$-fat $H$ minor is quasi-isometric to a graph with no $H$ minor. Our counterexample is furthermore not quasi-isometric to a graph with no 2-fat $H$ minor or a length space with no $H$ minor. On the other hand, we show that the following weakening holds: any graph with no $K$-fat $H$ minor is quasi-isometric to a graph with no $3$-fat $H$ minor.

math.CO↗

arXiv2024

Excluding a clique or a biclique in graphs of bounded induced matching treewidth

For a tree decomposition $\mathcal{T}$ of a graph $G$, let $\mu(\mathcal{T})$ denote the maximum size of an induced matching in $G$ with the property that some bag of $\mathcal{T}$ contains at least one endpoint of every edge of the matching. The induced matching treewidth of a graph $G$ is the minimum value of $\mu(\mathcal{T})$ over all tree decompositions $\mathcal{T}$ of $G$. Classes of graphs with bounded induced matching treewidth admit polynomial-time algorithms for a number of problems, including INDEPENDENT SET, $k$-COLORING, ODD CYCLE TRANSVERSAL, and FEEDBACK VERTEX SET. In this paper, we focus on combinatorial properties of such classes. First, we show that graphs with bounded induced matching treewidth that exclude a fixed biclique as an induced subgraph have bounded tree-independence number, which is another well-studied parameter defined in terms of tree decompositions. This sufficient condition about excluding a biclique is also necessary, as bicliques have unbounded tree-independence number. Second, we show that graphs with bounded induced matching treewidth that exclude a fixed clique have bounded chromatic number, that is, classes of graphs with bounded induced matching treewidth are $\chi$-bounded. The two results confirm two conjectures due to Lima et al. [ESA 2024].

math.CO↗

arXiv2024

On classes of bounded tree rank, their interpretations, and efficient sparsification

Graph classes of bounded tree rank were introduced recently in the context of the model checking problem for first-order logic of graphs. These graph classes are a common generalization of graph classes of bounded degree and bounded treedepth, and they are a special case of graph classes of bounded expansion. We introduce a notion of decomposition for these classes and show that these decompositions can be efficiently computed. Also, a natural extension of our decomposition leads to a new characterization and decomposition for graph classes of bounded expansion (and an efficient algorithm computing this decomposition). We then focus on interpretations of graph classes of bounded tree rank. We give a characterization of graph classes interpretable in graph classes of tree rank $2$. Importantly, our characterization leads to an efficient sparsification procedure: For any graph class $C$ interpretable in a efficiently bounded graph class of tree rank at most $2$, there is a polynomial time algorithm that to any $G \in C$ computes a (sparse) graph $H$ from a fixed graph class of tree rank at most $2$ such that $G = I(H)$ for a fixed interpretation $I$. To the best of our knowledge, this is the first efficient "interpretation reversal" result that generalizes the result of Gajarsk\'y et al. [LICS 2016], who showed an analogous result for graph classes interpretable in classes of graphs of bounded degree.

cs.DM↗

arXiv2024

A survey of degree-boundedness

Suppose a graph has no large balanced bicliques, but has large minimum degree. Then what can we say about its induced subgraphs? This question motivates the study of degree-boundedness, which is like $\chi$-boundedness but for minimum degree instead of chromatic number. We survey this area with an eye towards open problems.

math.CO↗

arXiv2023

First-Order Model Checking on Monadically Stable Graph Classes

A graph class $\mathscr{C}$ is called monadically stable if one cannot interpret, in first-order logic, arbitrary large linear orders in colored graphs from $\mathscr{C}$. We prove that the model checking problem for first-order logic is fixed-parameter tractable on every monadically stable graph class. This extends the results of [Grohe, Kreutzer, and Siebertz; J. ACM '17] for nowhere dense classes and of [Dreier, M\"ahlmann, and Siebertz; STOC '23] for structurally nowhere dense classes to all monadically stable classes. As a complementary hardness result, we prove that for every hereditary graph class $\mathscr{C}$ that is edge-stable (excludes some half-graph as a semi-induced subgraph) but not monadically stable, first-order model checking is $\mathrm{AW}[*]$-hard on $\mathscr{C}$, and $\mathrm{W}[1]$-hard when restricted to existential sentences. This confirms, in the special case of edge-stable classes, an on-going conjecture that the notion of monadic NIP delimits the tractability of first-order model checking on hereditary classes of graphs. For our tractability result, we first prove that monadically stable graph classes have almost linear neighborhood complexity. Using this, we construct sparse neighborhood covers for monadically stable classes, which provides the missing ingredient for the algorithm of [Dreier, M\"ahlmann, and Siebertz; STOC '23]. The key component of this construction is the usage of orders with low crossing number [Welzl; SoCG '88], a tool from the area of range queries. For our hardness result, we prove a new characterization of monadically stable graph classes in terms of forbidden induced subgraphs. We then use this characterization to show that in hereditary classes that are edge-stable but not monadically stable, one can effectively interpret the class of all graphs using only existential formulas.

cs.LO↗

arXiv2023

Universality for graphs with bounded density

A graph $G$ is $\textit{universal}$ for a (finite) family $\mathcal{H}$ of graphs if every $H \in \mathcal{H}$ is a subgraph of $G$. For a given family $\mathcal{H}$, the goal is to determine the smallest number of edges an $\mathcal{H}$-universal graph can have. With the aim of unifying a number of recent results, we consider a family of graphs with bounded density. In particular, we construct a graph with $O_d\left( n^{2 - 1/(\lceil d \rceil + 1)} \right)$ edges which contains every $n$-vertex graph with density at most $d \in \mathbb{Q}$ ($d \ge 1$), which is close to a lower bound $\Omega(n^{2 - 1/d - o(1)})$ obtained by counting lifts of a carefully chosen (small) graph. When restricting the maximum degree of such graphs to be constant, we obtain a near-optimal universality. If we further assume $d \in \mathbb{N}$, we get an asymptotically optimal construction.

math.CO↗