arXiv subjects

Prosenjit Bose

Publications and source records attributed to Prosenjit Bose.

At least 19 recordsLinked to original sources

arXiv2026

Containment Problems in Convex Sets

Let $P$ be a set of $n$ points in the plane, and let $C$ be a compact, convex subset of the plane that contains the origin in its interior. We study the problem of finding two points $p$ and $q$ of $P$ with the property that any homothet of $C$ that contains $p$ and $q$ must contain at least a constant fraction of the points of $P$. We give proofs for several cases. When $C$ is strictly convex we achieve a fraction of $1/5$, and when $C$ is a general convex region we achieve a fraction of $1/6e$. We show how the ideas of this second bound can be used to prove a lower bound for a version of the problem in $\mathbb{R}^d$ when $C$ is a $d$-ball. We also consider a few special cases. When $C$ is a triangle we get a lower bound of $1/12$, and when we allow that triangle to be rotated by $π$ radians in addition to being scaled and translated we get a lower bound of $1/768$. In proving this last result, we improve the state of the art on the version of this problem in three dimensions, where $C$ is a parallelepiped. We also consider the algorithmic aspects of finding such a pair of points.

cs.CG↗

arXiv2026

Connected Dominating Sets in Triangulations

A dominating set of a graph $G$ is \emph{connected} if it induces a connected graph in $G$. For planar triangulations, it has been known since 1990 that every $n$-vertex triangulation with $n\ge4$ admits a connected dominating set of size at most $n/2 - 1$, and no improvement to this bound was known for over three decades. We break this longstanding barrier by showing that every $n$-vertex triangulation has a connected dominating set of size at most $10n/21$. Equivalently, every triangulation admits a spanning tree with at least $11n/21$ leaves. This result has graph drawing applications, establishing a bound for one-bend free sets and improving the known bound for simultaneous planar embeddings. The connected dominating set results extend readily to show that an $n$-vertex triangulation of a surface of Euler genus $g$ has a connected dominating set of size at most $10n/21+O(\sqrt{gn})$.

math.CO↗

arXiv2026

Polychromatic 2-colorings with Bounded Discrepancy for Triangulations

A polychromatic $2$-coloring of a triangulation is a $2$-coloring of the vertices such that no face is monochromatic. The discrepancy of a coloring is the maximum difference between the sizes of the color classes. Asayama and Matsumoto (Graphs and Combinatorics, 2022) proved that every triangulation admits a polychromatic $2$-coloring with discrepancy at most $\tfrac{5n-16}{9}$, and that there exists a class of triangulations for which every polychromatic $2$-coloring has discrepancy at least $\tfrac{n}{3} - 2$, where $n$ is the number of vertices. We improve the upper bound, showing that every triangulation admits a polychromatic $2$-coloring with discrepancy at most $\tfrac{3n-16}{7}$ and such a $2$-coloring can be computed in quadratic time. We also show a discrepancy of at most $n-\tfrac{4M}{3}$ for triangulations with a matching of size $M$. This implies, for example, that Delaunay triangulations admit a discrepancy of at most $\tfrac{n}{3}$. We provide a linear-time algorithm to compute a $2$-coloring whose discrepancy is at most $\tfrac{5n-24}{7}$. One of our results shows that any proper four coloring with the largest color class of size $\frac{n}{2}$ would imply a $2$-coloring with discrepancy at most $\frac{n}{3}$. The existence of such a proper coloring has been recently confirmed by Kawarabayashi, Yoneda, and Yoneda (arXiv 2026). Therefore the two results together confirm the discrepancy of at most $\frac{n}{3}$ for triangulations.

cs.CG↗

arXiv2026

A Linear-Time Algorithm for Product Structure in Planar Graphs

The \emph{Product Structure Theorem} for planar graphs (Dujmović et al.\ \emph{JACM}, \textbf{67}(4):22) states that any planar graph is contained in the strong product of a planar $3$-tree, a path, and a $3$-cycle. We give a simple linear-time algorithm for finding this partition as well as several related partitions. This improves on the previous $O(n\log n)$ time algorithm for finding this partition (Morin.\ \emph{Algorithmica}, \textbf{85}(5):1544--1558). The algorithm is practical; we provide open-source C and C++ implementations that, even on modest hardware, can process graphs with millions of vertices in a matter of seconds.

cs.DS↗

arXiv2026

On the Spanning Ratio of the Greedy Triangulation for Convex Point Sets

The greedy triangulation of a finite planar point set is obtained by considering all segments in nondecreasing order of length and inserting each segment that does not cross an earlier one. Its spanning ratio is known to be bounded by a universal constant, but the standard bound obtained from the diamond and good-polygon properties is about $11739.1$. We prove a substantially smaller bound for points in convex position. In particular, for every finite point set $P\subset\mathbb{R}^2$ in convex position and every pair $u,v\in P$, the greedy triangulation contains a $u$--$v$ path of length at most $κ|uv|$, where $κ<17.814$. Thus, the greedy triangulation of a convex point set is an $18$-spanner.

cs.CG↗

arXiv2026

3-packings in Triangulations: Algorithms, bounds, and Complexity

We study $H$-packings in plane triangulations for the three-vertex graphs $H\in\{P_3,K_3,P_2\cup P_1\}$. For a graph $H$, let $λ_H(G)$ denote the maximum size of an $H$-packing in $G$, with the convention that for $H=P_2\cup P_1$ the copies are required to be induced. For $P_3$-packings, we prove that every triangulation $G$ on $n$ vertices satisfies $λ_{P_3}(G)\ge \left\lfloor \frac n5\right\rfloor$, and show that this lower bound is asymptotically tight. We also study triangle packings in triangulations and provide lower bounds for $λ_{K_3}(G)$ in terms of the maximum degree and the degree sequence. We give a face-path characterization of triangle factors in $4$-connected plane triangulations using a hamiltonian cycle and the weak duals of the two associated maximal outerplanar graphs. Finally, for induced packings by $P_2\cup P_1$, we prove that every plane triangulation $T$ on $n$ vertices satisfies $λ_{P_2\cup P_1}(T)\ge \left\lfloor \frac n3\right\rfloor-2$, and show that such a packing can be found in polynomial time.

cs.DM↗

arXiv2026

Closest Pair Queries in Vertical Slabs and Tight Bounds on the Number of Possible Answers

Let $S$ be a set of $n$ points in $\mathbb{R}^d$, where $d \geq 2$ is a constant, and let $H_1,H_2,\ldots,H_{m+1}$ be a sequence of vertical hyperplanes that are sorted by their first coordinates, such that exactly $n/m$ points of $S$ are between any two successive hyperplanes. Let $A(S,m)$ be the set of different closest pairs in the ${{m+1} \choose 2}$ vertical slabs that are bounded by $H_i$ and $H_j$, over all $1 \leq i < j \leq m+1$. We prove tight bounds for the largest possible size of $A(S,m)$, over all point sets of size $n$, and for all values of $1 \leq m \leq n$. As a result of these bounds, we obtain, for any constant $ε>0$, a data structure of size $O(n)$, such that for any vertical query slab $Q$, the closest pair in the set $Q \cap S$ can be reported in $O(n^{1/2+ε})$ time. Prior to this work, no linear space data structure with sublinear query time was known.

cs.CG↗

arXiv2026

On the Doubling Dimension and the Perimeter of Geodesically Convex Sets in Fat Polygons

Many algorithmic problems can be solved (almost) as efficiently in metric spaces of bounded doubling dimension as in Euclidean space. Unfortunately, the metric space defined by points in a simple polygon equipped with the geodesic distance does not necessarily have bounded doubling dimension. We therefore study the doubling dimension of fat polygons, for two well-known fatness definitions. We prove that locally-fat simple polygons do not always have bounded doubling dimension, while any $(α,β)$-covered polygon does have bounded doubling dimension (even if it has holes). We also study the perimeter of geodesically convex sets in $(α,β)$-covered polygons (possibly with holes), and show that this perimeter is at most a constant times the Euclidean diameter of the set. Using these two results, we obtain new results for several problems on $(α,β)$-covered polygons, including an algorithm that computes the closest pair of a set of $m$ points in an $(α,β)$-covered polygon with $n$ vertices that runs in $O(n + m\log{n})$ expected time.

cs.CG↗

arXiv2026

On geodesic disks enclosing many points

Let $ Π(n) $ be the largest number such that for every set $ S $ of $ n $ points in a polygon~$ P $, there always exist two points $ x, y \in S $, where every geodesic disk containing $ x $ and $ y $ contains $ Π(n) $ points of~$ S $. We establish upper and lower bounds for $ Π(n)$, and show that $ \left\lceil \frac{n}{5}\right\rceil+1 \leq Π(n) \leq \left\lceil \frac{n}{4} \right\rceil +1 $. We also show that there always exist two points $x, y\in S$ such that every geodesic disk with $x$ and $y$ on its boundary contains at least $ \frac{n}{7+\sqrt{37}} \approx \left\lceil \frac{n}{13.1} \right\rceil$ points both inside and outside the disk. For the special case where the points of $ S $ are restricted to be the vertices of a geodesically convex polygon we give a tight bound of $\left\lceil \frac{n}{3} \right\rceil + 1$. We provide the same tight bound when we only consider geodesic disks having $ x $ and $ y $ as diametral endpoints. We give upper and lower bounds of $\left\lceil \frac{n}{5} \right\rceil + 1 $ and $\frac{n}{6+\sqrt{26}} \approx \left\lceil \frac{n}{11.1} \right\rceil$, respectively, for the two-colored version of the problem. Finally, for the two-colored variant we show that there always exist two points $x, y\in S$ where $x$ and $y$ have different colors and every geodesic disk with $x$ and $y$ on its boundary contains at least $\left\lceil \frac{n}{27.1}\right\rceil+1$ points both inside and outside the disk.

cs.CG↗

arXiv2026

The Spanning Ratio of the Directed $Θ_6$-Graph is 5

Given a finite set $P\subset\mathbb{R}^2$, the directed Theta-6 graph, denoted $\vecΘ_6(P)$, is a well-studied geometric graph due to its close relationship with the Delaunay triangulation. The $\vecΘ_6(P)$-graph is defined as follows: the plane around each point $u\in P$ is partitioned into $6$ equiangular cones with apex $u$, and in each cone, $u$ is joined to the point whose projection on the bisector of the cone is closest. Equivalently, the $\vecΘ_6(P)$-graph contains an edge from $u$ to $v$ exactly when the interior of $\nabla_u^v$ is disjoint from $P$, where $\nabla_u^v$ is the unique equilateral triangle containing $u$ on a corner, $v$ on the opposite side, and whose sides are parallel to the cone boundaries. It was previously shown that the spanning ratio of the $\vecΘ_6(P)$-graph is between $4$ and $7$ in the worst case (Akitaya, Biniaz, and Bose \emph{Comput. Geom.}, 105-106:101881, 2022). We close this gap by showing a tight spanning ratio of 5. This is the first tight bound proven for the spanning ratio of any $\vecΘ_k(P)$-graph. Our lower bound models a long path by mapping it to a converging series. Our upper bound proof uses techniques novel to the area of spanners. We use linear programming to prove that among several candidate paths, there exists a path satisfying our bound.

cs.CG↗

arXiv2026

Cops and robber in graphs with bounded vertex cover number

Meyniel's conjecture states that $n$-vertex connected graphs have cop number $O(\sqrt{n})$. The current best known upper bound is $n/2^{(1-o(1))\sqrt{\log n}}$, proved independently by Lu and Peng (2011), and by Scott and Sudakov (2011). In this paper, we extend their result by showing that every connected graph with vertex cover number $k$ has cop number at most $k/2^{(1-o(1))\sqrt{\log k}}$. This is the first sublinear upper bound on the cop number in terms of the vertex cover number.

math.CO↗

arXiv2026

Computational aspects of disks enclosing many points

Let $S$ be a set of $n$ points in the plane. We present several different algorithms for finding a pair of points in $S$ such that any disk that contains that pair must contain at least $cn$ points of $S$, for some constant $c>0$. The first is a randomized algorithm that finds a pair in $O(n\log n)$ expected time for points in general position, and $c = 1/2-\sqrt{(1+2α)/12}$, for any $0<α<1$. The second algorithm, also for points in general position, takes quadratic time, but the constant $c$ is improved to $1/2-1/{\sqrt{12}} \approx 1/4.7$. The second algorithm can also be used as a subroutine to find the pair that maximizes the number of points inside any disk that contains the pair, in $O(n^2\log n)$ time. We also consider variants of the problem. When the set $S$ is in convex position, we present an algorithm that finds in linear time a pair of points such that any disk through them contains at least $n/3$ points of $ S $. For the variant where we are only interested in finding a pair such that the diametral disk of that pair contains many points, we also have a linear-time algorithm that finds a disk with at least $n/3$ points of $S$. Finally, we present a generalization of the first two algorithms to the case where the set $S$ of points is coloured using two colours. We also consider adapting these algorithms to solve the same problems when $S$ is a set of points inside of a simple polygon $P$, with the notion of a disk replaced by that of a geodesic disk.

cs.CG↗

arXiv2025

Noncrossing Longest Paths and Cycles

Edge crossings in geometric graphs are sometimes undesirable as they could lead to unwanted situations such as collisions in motion planning and inconsistency in VLSI layout. Short geometric structures such as shortest perfect matchings, shortest spanning trees, shortest spanning paths, and shortest spanning cycles on a given point set are inherently noncrossing. However, the longest such structures need not be noncrossing. In fact, it is intuitive to expect many edge crossings in various geometric graphs that are longest. Recently, Álvarez-Rebollar, Cravioto-Lagos, Marín, Solé-Pi, and Urrutia (Graphs and Combinatorics, 2024) constructed a set of points for which the longest perfect matching is noncrossing. They raised several challenging questions in this direction. In particular, they asked whether the longest spanning path, on any finite set of points in the plane, must have a pair of crossing edges. They also conjectured that the longest spanning cycle must have a pair of crossing edges. In this paper, we give a negative answer to the question and also refute the conjecture. We present a framework for constructing arbitrarily large point sets for which the longest perfect matchings, the longest spanning paths, and the longest spanning cycles are noncrossing.

cs.CG↗

arXiv2025

Computing Vertex and Edge Connectivity of Graphs Embedded with Crossings

Vertex connectivity and edge connectivity are fundamental concepts in graph theory that have been widely studied from both structural and algorithmic perspectives. The focus of this paper is on computing these two parameters for graphs embedded on the plane with crossings. For planar graphs -- which can be embedded on the plane without any crossings -- it has long been known that vertex and edge connectivity can be computed in linear time. Recently, the algorithm for vertex connectivity was extended from planar graphs to 1-plane graphs (where each edge is crossed at most once) without $\times$-crossings -- these are crossings whose endpoints induce a matching. The key insight, for both these classes of graphs, is that any two vertices/edges of a minimum vertex/edge cut have small face-distance (distance measured by number of faces) in the embedding. In this paper, we attempt at a comprehensive generalization of this idea to a wider class of graphs embedded on the plane. Our method works for all those embedded graphs where every pair of crossing edges is connected by a path whose vertices and edges have a small face-distance from the crossing point. Important examples of such graphs include optimal 2-planar and optimal 3-planar graphs, $d$-map graphs, $d$-framed graphs, graphs with bounded crossing number, and $k$-plane graphs with bounded number of $\times$-crossings. For all these graph classes, we get a linear-time algorithm for computing vertex and edge connectivity.

cs.DS↗

arXiv2025

An Improved Bound for Plane Covering Paths

A covering path for a finite set $P$ of points in the plane is a polygonal path such that every point of $P$ lies on a segment of the path. The vertices of the path need not be at points of $P$. A covering path is plane if its segments do not cross each other. Let $π(n)$ be the minimum number such that every set of $n$ points in the plane admits a plane covering path with at most $π(n)$ segments. We prove that $π(n)\le \lceil6n/7\rceil$. This improves the previous best-known upper bound of $\lceil 21n/22\rceil$, due to Biniaz (SoCG 2023). Our proof is constructive and yields a simple $O(n\log n)$-time algorithm for computing a plane covering path.

cs.CG↗

arXiv2025

Cops and Robbers for Graphs on Surfaces with Crossings

Cops and Robbers is a game played on a graph where a set of cops attempt to capture a single robber. The game proceeds in rounds, where each round first consists of the cops' turn, followed by the robber's turn. In the cops' turn, every cop can choose to either stay on the same vertex or move to an adjacent vertex, and likewise the robber in his turn. The robber is considered to be captured if, at any point in time, there is some cop on the same vertex as the robber. A natural question in this game concerns the cop-number of a graph -- the minimum number of cops needed to capture the robber. It has long been known that graphs embeddable (without crossings) on surfaces of bounded genus have bounded cop-number. In contrast, the class of 1-planar graphs -- graphs that can be drawn on the plane with at most one crossing per edge -- does not have bounded cop-number. This paper initiates an investigation into how distance between crossing pairs of edges influences a graph's cop number. In particular, we look at Distance $d$ Cops and Robbers, a variant of the classical game, where the robber is considered to be captured if there is a cop within distance $d$ of the robber. Let $c_d(G)$ denote the minimum number of cops required in the graph $G$ to capture a robber within distance $d$. We look at various classes of graphs, such as 1-plane graphs, $k$-plane graphs (graphs where each edge is crossed at most $k$ times), and even general graph drawings, and show that if every crossing pair of edges can be connected by a path of small length, then $c_d(G)$ is bounded, for small values of $d$.

cs.DM↗

arXiv2025

Tight Routing and Spanning Ratios of Arbitrary Triangle Delaunay Graphs

A Delaunay graph built on a planar point set has an edge between two vertices when there exists a disk with the two vertices on its boundary and no vertices in its interior. When the disk is replaced with an equilateral triangle, the resulting graph is known as a Triangle-Distance Delaunay Graph or TD-Delaunay for short. A generalized $\text{TD}_{θ_1,θ_2}$-Delaunay graph is a TD-Delaunay graph whose empty region is a scaled translate of a triangle with angles of $θ_1,θ_2,θ_3:=π-θ_1-θ_2$ with $θ_1\leqθ_2\leqθ_3$. We prove that $\frac{1}{\sin(θ_1/2)}$ is a lower bound on the spanning ratio of these graphs which matches the best known upper bound (Lubiw & Mondal, J. Graph Algorithms Appl., 23(2):345-369). Then we provide an online local routing algorithm for $\text{TD}_{θ_1,θ_2}$-Delaunay graphs with a routing ratio that is optimal in the worst case. When $θ_1=θ_2=\fracπ{3}$, our expressions for the spanning ratio and routing ratio evaluate to $2$ and $\frac{\sqrt{5}}{3}$, matching the known tight bounds for TD-Delaunay graphs.

cs.CG↗

arXiv2025

On Separating Path and Tree Systems in Graphs

We explore the concept of separating systems of vertex sets of graphs. A separating system of a set $X$ is a collection of subsets of $X$ such that for any pair of distinct elements in $X$, there exists a set in the separating system that contains exactly one of the two elements. A separating system of the vertex set of a graph $G$ is called a vertex-separating path (tree) system of $G$ if the elements of the separating system are paths (trees) in the graph $G$. In this paper, we focus on the size of the smallest vertex-separating path (tree) system for different types of graphs, including trees, grids, and maximal outerplanar graphs.

cs.DM↗