arXiv subjects

Morteza Saghafian

Publications and source records attributed to Morteza Saghafian.

At least 19 recordsLinked to original sources

arXiv2026

Lunar Generalizations of the Euclidean Minimum Spanning Tree in the Plane and their Expected Costs

Motivated by the recent introduction of chromatic persistent homology, we generalize the Euclidean minimum spanning tree (EMST) for $n$ points in $\mathbb{R}^2$ to the lunar EMST for the case in which the points come in $s+1$ colors. Calling the intersection of $s+1$ disks of radius $r$ centered at points with pairwise different colors a \emph{lune}, the generalized EMST reflects the history of the union of lunes as $r$ goes from $0$ to $\infty$, and its \emph{cost} is twice the difference between the radii when the arcs and nodes of the tree are formed. If the points are chosen uniformly at random in $[0,1]^2$ and colored randomly, the expected cost converges to some constant (that depends on $s$) times $\sqrt{n}$, as $n$ goes to infinity. The main contribution of this paper is a proof that this constant exists, however similar to the case of the classic EMST, its precise value remains elusive.

cs.CG↗

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

CurveBench: A Benchmark for Exact Topological Reasoning over Nested Jordan Curves

We introduce CurveBench, a benchmark for hierarchical topological reasoning from visual input. CurveBench consists of \textbf{756 images} of pairwise non-intersecting Jordan curves across easy, polygonal, topographic-inspired, maze-like, and dense counting configurations. Each image is annotated with a rooted tree encoding the containment relations between planar regions. We formulate the task as structured prediction: given an image, a model must recover the full rooted containment tree induced by the curves. Despite the visual simplicity of the task, the strongest evaluated model, Gemini 3.1 Pro, achieves only \textbf{71.1\%} tree-generation accuracy on CurveBench-Easy and \textbf{19.1\%} on CurveBench-Hard. We further demonstrate benchmark utility through RLVR-style fine-tuning of open-weight vision-language models. Our trained Qwen3-VL-8B model improves over \texttt{Qwen-3-VL-8B-Thinking} from \textbf{2.8\%} to \textbf{33.3\%} tree-generation accuracy on CurveBench-Easy, exceeding GPT-5.4 and Claude Opus 4.5 under our evaluation protocol. The remaining gap, especially on CurveBench-Hard, shows that exact topology-aware visual reasoning remains far from solved.

cs.CV↗

arXiv2026

Covering Complete Geometric Graphs by Monotone Paths

Given a set $A$ of $n$ points (vertices) in general position in the plane, the \emph{complete geometric graph} $K_n[A]$ consists of all $\binom{n}{2}$ segments (edges) between the elements of $A$. It is known that the edge set of every complete geometric graph on $n$ vertices can be partitioned into $O(n^{3/2})$ crossing-free paths (or matchings). We strengthen this result under various additional assumptions on the point set. In particular, we prove that for a set $A$ of $n$ \emph{randomly} selected points, uniformly distributed in $[0,1]^2$, with probability tending to $1$ as $n\rightarrow\infty$, the edge set of $K_n[A]$ can be covered by $O(n\log n)$ crossing-free paths and by $O(n\sqrt{\log n})$ crossing-free matchings. On the other hand, we construct $n$-element point sets such that covering the edge set of $K_n[A]$ requires a quadratic number of monotone paths.

math.CO↗

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

Expected Length of the Euclidean Minimum Spanning Tree and 1-norms of Chromatic Persistence Diagrams in the Plane

Let $c$ be the constant such that the expected length of the Euclidean minimum spanning tree of $n$ random points in the unit square is $c \sqrt{n}$ in the limit, when $n$ goes to infinity. We improve the prior best lower bound of $0.6008 \leq c$ by Avram and Bertsimas to $0.6289 \leq c$. The proof is a by-product of studying the persistent homology of randomly $2$-colored point sets. Specifically, we consider the filtration induced by the inclusions of the two mono-chromatic sublevel sets of the Euclidean distance function into the bi-chromatic sublevel set of that function. Assigning colors randomly, and with equal probability, we show that the expected $1$-norm of each chromatic persistence diagram is a constant times $\sqrt{n}$ in the limit, and we determine the constant in terms of $c$ and another constant, $c_L$, which arises for a novel type of Euclidean minimum spanning tree of $2$-colored point sets.

math.PR↗

arXiv2025

Flips in Two-dimensional Hypertriangulations

We study flips in hypertriangulations of planar points sets. Here a level-$k$ hypertriangulation of $n$ points in the planes is a subdivision induced by the projection of a $k$-hypersimplex, which is the convex hull of the barycenters of the $(k-1)$-dimensional faces of the standard $(n-1)$-simplex. In particular, we introduce four types of flips and prove that the level-2 hypertriangulations are connected by these flips.

math.CO↗

arXiv2025

On Spheres with $k$ Points Inside

We generalize the classic definition of Delaunay triangulation and prove that for a locally finite and coarsely dense generic point set, $A \subseteq \mathbb{R}^d$, the $d$-simplices whose vertices belong to $A$ and whose circumscribed spheres enclose exactly $k$ points of $A$ cover $\mathbb{R}^d$ exactly $\binom{d+k}{d}$ times. Similarly, the subset of such simplices incident to a point in $A$ cover any small enough neighborhood of that point exactly $\binom{d+k-1}{d-1}$ times. We extend this result to the cases in which the points are weighted and when $A$ contains only finitely many points in $\mathbb{R}^d$ or in $\mathbb{S}^d$. Using these results, we give new proofs of classic results on $k$-facets, old and new combinatorial results for hyperplane arrangements, and a new proof for the fact that the volumes of hypersimplices are Eulerian numbers.

math.CO↗

arXiv2025

On the MST-ratio: Theoretical Bounds and Complexity of Finding the Maximum

Given a finite set of red and blue points in $\Rspace^d$, the MST-ratio is defined as the total length of the Euclidean minimum spanning trees of the red points and the blue points, divided by the length of the Euclidean minimum spanning tree of their union. The MST-ratio has recently gained attention due to its direct interpretation in topological models for studying point sets with applications in spatial biology. The maximum MST-ratio of a point set is the maximum MST-ratio over all proper colorings of its points by red and blue. We prove that finding the maximum MST-ratio of a given point set is NP-hard when the dimension is part of the input. Moreover, we present a quadratic-time $3$-approximation algorithm for this problem. As part of the proof, we show that, in any metric space, the maximum MST-ratio is smaller than $3$. Additionally, we study the average MST-ratio over all colorings of a set of $n$ points. We show that this average is always at least $\frac{n-2}{n-1}$, and for $n$ random points uniformly distributed in a $d$-dimensional unit cube, the average tends to $\sqrt[d]{2}$ in expectation as $n$ approaches infinity.

cs.CG↗

arXiv2024

Order-2 Delaunay Triangulations Optimize Angles

The local angle property of the (order-$1$) Delaunay triangulations of a generic set in $\mathbb{R}^2$ asserts that the sum of two angles opposite a common edge is less than $π$. This paper extends this property to higher order and uses it to generalize two classic properties from order-$1$ to order-$2$: (1) among the complete level-$2$ hypertriangulations of a generic point set in $\mathbb{R}^2$, the order-$2$ Delaunay triangulation lexicographically maximizes the sorted angle vector; (2) among the maximal level-$2$ hypertriangulations of a generic point set in $\mathbb{R}^2$, the order-$2$ Delaunay triangulation is the only one that has the local angle property. We also use our method of establishing (2) to give a new short proof of the angle vector optimality for the (order-1) Delaunay triangulation. For order-$1$, both properties have been instrumental in numerous applications of Delaunay triangulations, and we expect that their generalization will make order-$2$ Delaunay triangulations more attractive to applications as well.

math.CO↗

arXiv2024

The Euclidean MST-ratio for Bi-colored Lattices

Given a finite set, $A \subseteq \mathbb{R}^2$, and a subset, $B \subseteq A$, the \emph{MST-ratio} is the combined length of the minimum spanning trees of $B$ and $A \setminus B$ divided by the length of the minimum spanning tree of $A$. The question of the supremum, over all sets $A$, of the maximum, over all subsets $B$, is related to the Steiner ratio, and we prove this sup-max is between $2.154$ and $2.427$. Restricting ourselves to $2$-dimensional lattices, we prove that the sup-max is $2.0$, while the inf-max is $1.25$. By some margin the most difficult of these results is the upper bound for the inf-max, which we prove by showing that the hexagonal lattice cannot have MST-ratio larger than $1.25$.

cs.CG↗

arXiv2024

Chromatic Alpha Complexes

Motivated by applications in the medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective. Based on the persistent homology for images, kernels and cokernels, we design provably stable homological quantifiers that describe the geometric micro- and macro-structure of how the color classes mingle. These can be efficiently computed using chromatic variants of Delaunay and alpha complexes, and code that does these computations is provided.

math.AT↗

arXiv2024

Chromatic Topological Data Analysis

Exploring the shape of point configurations has been a key driver in the evolution of TDA (short for topological data analysis) since its infancy. This survey illustrates the recent efforts to broaden these ideas to model spatial interactions among multiple configurations, each distinguished by a color. It describes advances in this area and prepares the ground for further exploration by mentioning unresolved questions and promising research avenues while focusing on the overlap with discrete geometry.

cs.CG↗

arXiv2024

Brillouin Zones of Integer Lattices and Their Perturbations

For a locally finite set, $A \subseteq \mathbb{R}^d$, the $k$-th Brillouin zone of $a \in A$ is the region of points $x \in \mathbb{R}^d$ for which $\|x-a\|$ is the $k$-th smallest among the Euclidean distances between $x$ and the points in $A$. If $A$ is a lattice, the $k$-th Brillouin zones of the points in $A$ are translates of each other, which tile space. Depending on the value of $k$, they express medium- or long-range order in the set. We study fundamental geometric and combinatorial properties of Brillouin zones, focusing on the integer lattice and its perturbations. Our results include the stability of a Brillouin zone under perturbations, a linear upper bound on the number of chambers in a zone for lattices in $\mathbb{R}^2$, and the convergence of the maximum volume of a chamber to zero for the integer lattice.

math.CO↗

arXiv2024

Approximating Simplet Frequency Distribution for Simplicial Complexes

Simplets, constituting elementary units within simplicial complexes (SCs), serve as foundational elements for the structural analysis of SCs. Previous efforts have focused on the exact count or approximation of simplet count rather than their frequencies, with the latter being more practical in large-scale SCs. This paper enables simplet frequency analysis of SCs by introducing the Simplet Frequency Distribution (SFD) vector. In addition, we present a bound on the sample complexity required for accurately approximating the SFD vector by any uniform sampling-based algorithm. We also present a simple algorithm for this purpose and justify the theoretical bounds with experiments on some random simplicial complexes.

cs.CG↗

arXiv2024

Grid Peeling of Parabolas

Grid peeling is the process of repeatedly removing the convex hull vertices of the grid-points that lie inside a given convex curve. It has been conjectured that, for a more and more refined grid, grid peeling converges to a continuous process, the affine curve-shortening flow, which deforms the curve based on the curvature. We prove this conjecture for one class of curves, parabolas with a vertical axis, and we determine the value of the constant factor in the formula that relates the two processes.

cs.CG↗

arXiv2023

Decomposition of Geometric Graphs into Star Forests

We solve a problem of Dujmović and Wood (2007) by showing that a complete convex geometric graph on $n$ vertices cannot be decomposed into fewer than $n-1$ star-forests, each consisting of noncrossing edges. This bound is clearly tight. We also discuss similar questions for abstract graphs.

math.CO↗