arXiv subjects

Todor Antić

Publications and source records attributed to Todor Antić.

10 recordsLinked to original sources

arXiv2026

How many times can two minimum spanning trees cross?

Let $P$ be a generic set of $n$ points in the plane, and let $P=R\cup B$ be a coloring of $P$ in two colors. We are interested in the number of crossings between the minimum spanning trees (MSTs) of $R$ and $B$, denoted by $\crossAB(R,B)$. We define the \emph{bicolored MST crossing number} of $P$, denoted by $\cross(P)$, as $\cross(P) = \max_{P= R\cup B}(\crossAB(R,B))$. We prove a linear upper bound for $\cross(P)$ when $P$ is generic. If $P$ is dense or in convex position, we provide linear lower bounds. Lastly, if $P$ is chosen uniformly at random from the unit square and is colored uniformly at random, we prove that the expected value of $\crossAB(R,B)$ is linear.

cs.CG↗

arXiv2026

How Close is a Tree to a Euclidean Minimum Spanning Tree?

Let $Γ$ be a straight-line crossing-free drawing of a tree $T$. A \emph{bad pair} in $Γ$ is a pair of non-adjacent vertices of $T$ whose Euclidean distance in $Γ$ is smaller than the length of the longest edge in the path connecting them in~$Γ$. When $Γ$ has no bad pairs, $Γ$ is a Euclidean Minimum Spanning Tree of its vertex set (or EMST-drawing for short). Deciding whether a tree of maximum degree at most six admits an EMST-drawing is known to be \NP-hard. In contrast, we characterize those caterpillars that admit an EMST-drawing. The characterization gives rise to a linear-time algorithm that decides if a caterpillar admits an EMST-drawing, and in the affirmative case, computes such a drawing. For caterpillars of maximum degree six, we further present a linear-time algorithm to compute a crossing-free straight-line drawing with the minimum number of bad pairs. For $n$-vertex trees with maximum vertex degree $Δ$, we prove the $Δ^2n\log n$ upper bound on the minimum number of bad pairs. In the special case of stars, we construct a drawing with the minimum number of bad pairs.

cs.CG↗

arXiv2026

QEDBENCH: Quantifying the Alignment Gap in Automated Evaluation of University-Level Mathematical Proofs

As Large Language Models (LLMs) saturate elementary benchmarks, the research frontier has shifted from generation to the reliability of automated evaluation. We demonstrate that standard "LLM-as-a-Judge" protocols suffer from a systematic Alignment Gap when applied to upper-undergraduate to early graduate level mathematics. To quantify this, we introduce QEDBench, the first large-scale dual-rubric alignment benchmark to systematically measure alignment with human experts on university-level math proofs by contrasting course-specific rubrics against expert common knowledge criteria. By deploying a dual-evaluation matrix (7 judges x 5 solvers) against 1,000+ hours of human evaluation, we reveal that certain frontier evaluators like Claude Opus 4.5, DeepSeek-V3, Qwen 2.5 Max, and Llama 4 Maverick exhibit significant positive bias (up to +0.18, +0.20, +0.30, +0.36 mean score inflation, respectively). Furthermore, we uncover a critical reasoning gap in the discrete domain: while Gemini 3.0 Pro achieves state-of-the-art performance (0.91 average human evaluation score), other reasoning models like GPT-5 Pro and Claude Sonnet 4.5 see their performance significantly degrade in discrete domains. Specifically, their average human evaluation scores drop to 0.72 and 0.63 in Discrete Math, and to 0.74 and 0.50 in Graph Theory. In addition to these research results, we also release QEDBench as a public benchmark for evaluating and improving AI judges. Our benchmark is publicly published at https://github.com/qqliu/Yale-QEDBench.

cs.LG↗

arXiv2026

Two Results on Outer-String Graphs

An \emph{outer-string representation} of a graph $G$ is an intersection representation of $G$ where vertices are represented by curves (strings) inside the unit disk and each curve has exactly one endpoint on the boundary of the unit disk (the anchor of the curve). Additionally, if each two curves are allowed to cross at most once, we call this an \emph{outer-$1$-string representation} of $G$. If we impose a cyclic ordering on the vertices of $G$ and require the cyclic order of the anchors to respect this cyclic order, such a representation is called a \emph{constrained outer-string representation}. In this paper, we present two results about graphs admitting outer-string representations. Firstly, we show that for a bipartite graph $G$ (and, more generally, for any $\{C_3,C_5\}$-free graph $G$) with a given cyclic order of vertices, we can decide in polynomial time whether $G$ admits a constrained outer-string representation. Our algorithm follows from a characterization by a single forbidden configuration, similar to that of Biedl et al. [GD 2024] for chordal graphs. Secondly, we answer an open question from the same authors and show that determining whether a given graph admits an outer-1-string representation is NP-hard. More generally, we show that it is NP-hard to determine if a given graph $G$ admits an outer-$k$-string representation for any fixed $k\ge1$.

math.CO↗

arXiv2026

Hypercube drawings with no long plane paths

We study the existence of plane substructures in drawings of the $d$-dimensional hypercube graph $Q_d$. We construct drawings of $Q_d$ which contain no plane subgraph with more than $2d-2$ edges, no plane path with more than $2d-3$ edges, and no plane matching of size more than $2d-4$. On the other hand, we prove that every rectilinear drawing of $Q_d$ with vertices in convex position contains a plane path of length $d$ (if $d$ is odd) or $d-1$ (if $d$ is even). We also prove that if a graph $G$ is a plane subgraph of every drawing of $Q_d$ for a sufficiently large $d$, then $G$ is necessarily a forest of caterpillars. Lastly, we give a short proof of a generalization of a result by Alpert et al. [Cong. Numerantium, 2009] on the maximum rectilinear crossing number of $Q_d$.

cs.CG↗

arXiv2025

Edge-Constrained Hamiltonian Paths on a Point Set

Let S be a set of distinct points in general position in the Euclidean plane. A plane Hamiltonian path on S is a crossing-free geometric path such that every point of S is a vertex of the path. It is known that, if S is sufficiently large, there exist three edge-disjoint plane Hamiltonian paths on S. In this paper we study an edge-constrained version of the problem of finding Hamiltonian paths on a point set. We first consider the problem of finding a single plane Hamiltonian path pi with endpoints s, t in S and constraints given by a segment ab, where a, b in S. We consider the following scenarios: (i) ab in pi; (ii) ab not in pi. We characterize those quintuples (S, a, b, s, t) for which pi exists. Secondly, we consider the problem of finding two plane Hamiltonian paths pi_1, pi_2 on a set S with constraints given by a segment ab, where a, b in S. We consider the following scenarios: (i) pi_1 and pi_2 share no edges and ab is an edge of pi_1; (ii) pi_1 and pi_2 share no edges and none of them includes ab as an edge; (iii) both pi_1 and pi_2 include ab as an edge and share no other edges. In all cases, we characterize those triples (S, a, b) for which pi_1 and pi_2 exist.

cs.CG↗

arXiv2025

Crossing and non-crossing families

For a finite set $P$ of points in the plane in general position, a \emph{crossing family} of size $k$ in $P$ is a collection of $k$ line segments with endpoints in $P$ that are pairwise crossing. It is a long-standing open problem to determine the largest size of a crossing family in any set of $n$ points in the plane in general position. It is widely believed that this size should be linear in $n$. Motivated by results from the theory of partitioning complete geometric graphs, we study a variant of this problem for point sets $P$ that do not contain a \emph{non-crossing family} of size $m$, which is a collection of 4 disjoint subsets $P_1$, $P_2$, $P_3$, and $P_4$ of $P$, each containing $m$ points of $P$, such that for every choice of 4 points $p_i \in P_i$, the set $\{p_1,p_2,p_3,p_4\}$ is such that $p_4$ is in the interior of the triangle formed by $p_1,p_2,p_3$. We prove that, for every $m \in \mathbb{N}$, each set $P$ of $n$ points in the plane in general position contains either a crossing family of size $n/2^{O(\sqrt{\log{m}})}$ or a non-crossing family of size $m$, by this strengthening a recent breakthrough result by Pach, Rubin, and Tardos (2021). Our proof is constructive and we show that these families can be obtained in expected time $O(nm^{1+o(1)})$. We also prove that a crossing family of size $Ω(n/m)$ or a non-crossing family of size $m$ in $P$ can be found in expected time $O(n)$.

math.CO↗

arXiv2025

Unbent Collections of Orthogonal Drawings

Recently, there has been interest in representing single graphs by multiple drawings; for example, using graph stories, storyplans, or uncrossed collections. In this paper, we apply this idea to orthogonal graph drawing. Due to the orthogonal drawing style, we focus on 4-graphs, that is, graphs of maximum degree 4. We restrict ourselves to plane graphs, that is, planar graphs whose embedding is fixed. Our goal is to represent any plane 4-graph $G$ by an unbent collection, that is, a collection of orthogonal drawings of $G$ that adhere to the embedding of $G$ and ensure that each edge of $G$ is drawn without bends in at least one of the drawings. We investigate two objectives. First, we consider minimizing the number of drawings in an unbent collection. We prove that every plane 4-graph can be represented by a collection with at most three drawings, which is tight. We also give necessary and sufficient conditions for a graph to admit an unbent collection of size $2$. Second, we consider minimizing the total number of bends over all drawings in an unbent collection. We show that this problem is NP-hard and give a 3-approximation algorithm. For the special case of plane triconnected cubic graphs, we show how to compute minimum-bend collections in linear time.

cs.CG↗

arXiv2024

Reconfigurations of Plane Caterpillars and Paths

Let $S$ be a point set in the plane, $\mathcal{P}(S)$ and $\mathcal{C}(S)$ sets of all plane spanning paths and caterpillars on $S$. We study reconfiguration operations on $\mathcal{P}(S)$ and $\mathcal{C}(S)$. In particular, we prove that all of the commonly studied reconfigurations on plane spanning trees still yield connected reconfiguration graphs for caterpillars when $S$ is in convex position. If $S$ is in general position, we show that the rotation, compatible flip and flip graphs of $\mathcal{C}(S)$ are connected while the slide graph is disconnected. For paths, we prove the existence of a connected component of size at least $2^{n-1}$ and that no component of size at most $7$ can exist in the flip graph on $\mathcal{P}(S)$.

math.CO↗

arXiv2024

Star-Forest Decompositions of Complete Graphs

We deal with the problem of decomposing a complete geometric graph into plane star-forests. In particular, we disprove a recent conjecture by Pach, Saghafian and Schnider by constructing for each $n$ a complete geometric graph on $n$ vertices which can be decomposed into $\frac{n}{2}+1$ plane star-forests. Additionally we prove that for even $n$, every decomposition of complete abstract graph on $n$ vertices into $\frac{n}{2}+1$ star-forests is composed of a perfect matching and $\frac{n}{2}$ star-forests with two edge-balanced components, which we call broken double stars.

math.CO↗