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Complement Submodular Information Measures for Balanced and Robust Data Selection A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs Laplacian Spectrum of the Weakly Zero-Divisor Graph of a Finite Commutative Ring An identity for second Eulerian numbers via lattice-point counting $t$-tone edge coloring of graphs Constructing Maximal Bumpless Pipedreams for Double Grothendieck Polynomials Mubayi's Polynomial-Ideal Conjecture and Cover-Ideal Turán Methods Implicit Binarization via Complex Phase Dynamics in Combinatorial Optimization The limits of Schur multipliers in Pólya conversion problems for the $q$-permanent function Universality theorems for generalized splines Framing Triangulations for Arbitrary Integer Flow Polytopes On the Common Generalization of Gentle Algebras and Framed Directed Acyclic Graphs The complexity of frugal digraph homomorphisms Chaotic and periodic behavior of jeu de taquin on infinite Young tableaux Enumerating Pattern Avoiding Parking Functions Incidence toric ideals and three-point functions Unique Winning Opening Move in Three-Row Chomp Strong majority colorings of graphs A Balancing Theorem for Spanning Trees of Rectangular Grid Graphs Spectral radius and edge-disjoint connected factors of graphs New invariants for rank metric codes, with applications to the classification of rank two semifields of order 256 Flexible DP-4-coloring of planar graphs without 4-cycles and intersecting triangles Balanced intersection size distributions in projective planes List Reconstruction Problem with List Size Two Is Dimensionality a Barrier for Retrieval Models? The INIEP: Irreducible and Positive Realizations The number of Pfaffian orientations on punctured polygonally cellulated surfaces Explicit Construction of Polytopes whose Ehrhart Polynomials Realize any Given Sign Pattern Finite-state enumeration of adjacency-constrained 132-avoiding permutations AMDS and quantum AMDS Constacyclic codes of length $4p^ς$ over $\mathbb{F}_{{p}^{m}}$
Vertex addition to a ball graph with application to relia...
Calum Buchanan, Puck Rombach, James Bagrow, Hamid R. Ossareh · 2025-06-24 · via math.CO updates on arXiv.org

A unit ball graph consists of a set of vertices, labeled by points in Euclidean space, and edges joining all pairs of points within distance 1. These geometric graphs are used to model a variety of spatial networks, including communication networks between agents in an autonomous swarm. In such an application, vertices and/or edges of the graph may not be perfectly reliable; an agent may experience failure or a communication link rendered inoperable. With the goal of designing robust swarm formations, or unit ball graphs with high reliability (probability of connectedness), in a preliminary conference paper we provided an algorithm with cubic time complexity to determine all possible changes to a unit ball graph by repositioning a single vertex. Using this algorithm and Monte Carlo simulations, one obtains an efficient method to modify a unit ball graph by moving a single vertex to a location which maximizes the reliability. Another important consideration in many swarm missions is area coverage, yet highly reliable ball graphs often contain clusters of vertices. Here, we generalize our previous algorithm to improve area coverage as well as reliability. Our algorithm determines a location to add or move a vertex within a unit ball graph which maximizes the reliability, under the constraint that no other vertices of the graph be within some fixed distance. We compare this method of obtaining graphs with high reliability and evenly distributed area coverage to another method which uses a modified Fruchterman-Reingold algorithm for ball graphs.