Let G be a complete edge-weighted graph on n vertices. To each subset of vertices of G assign the cost of the minimum spanning tree of the subset as its weight. Suppose that n is a multiple of some fixed positive integer k. The k-matching problem is the problem of finding a partition of the vertices of G into k-sets (sets of k elements), that minimizes the sum of the weights of the k-sets. The case of k = 3 has been shown to be NP-hard [Johnsson et al., 1998]. In the Euclidean version, the vertices of G are points in the plane and the weight of an edge is the Euclidean distance between its endpoints. We call this problem the Euclidean k-matching problem. We show that, for every fixed k >= 3, the Euclidean k-matching problem is NP-hard. This resolves an open problem in the literature and provides the first theoretical justification for the use of known heuristic methods in the case of k = 3. We also show that the problem remains NP-hard if the trees are required to be paths. (c) 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Let G be a graph on n vertices and 1 <= k <= n a fixed integer. The k-token graph of G is the graph Fk(G) whose vertex set consists of all k-subsets of the vertex set of G, where two vertices A and B are adjacent in Fk(G) whenever their symmetric difference A triangle B is an edge of G. In this paper we study the treewidth of Fk(G) when G is a star, path, or a complete graph. We show that in the first two cases, the treewidth is of order Theta(nk-1), and of order Theta(nk) in the third case. We conjecture that our upper bound for the treewidth of Fk(Kn) is tight. This is particularly relevant since Fk(Kn) is isomorphic to the well known Johnson graph J(n, k). (c) 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Order types are an equivalence relation between point configurations that capture their combinatorial and convexity properties. Let P be a κ-colored sequence of n ≥ d+1 points in general position in ℝ^d. Let ρ be a κ-colored order type on k ≤ d+1 points that has positive density on P; that is, for some constant δ>0, there are δ·nk k-point subsequences of P that have the same order type as ρ and the same color pattern. In this paper we show that there exists a constant c >0 (depending only on d, δ, k and κ) and disjoint subsets X_1,…,X_k of P, each with at least c · n points, such that for every choice of k points x_i ∈ X_i, (x_1,…,x_k) has the same order type and color pattern as ρ.

A rectilinear drawing of a graph is a drawing of the graph in the plane in which the edges are drawn as straight-line segments and its vertices are points in general position. The rectilinear crossing number of a graph is the minimum number of pairs of edges that cross over all rectilinear drawings of the graph. Let n >= r be positive integers. The graph K-n (R), is the complete balanced r-partite graph on n vertices, in which every set of the partition has at least & LeftFloor;n/r & RightFloor; vertices. The balanced layered graph, L-n (R), is an r-partite graph on n vertices, where n is multiple of r. Every partition of L-n (R) contains n/r vertices; for every 1 <= i <= r-1, all the vertices in the i-th partition are adjacent to all the vertices in the (i+1)-th partition, and these are the only edges of L-n (R). In this paper, we give upper bounds on the rectilinear crossing numbers of K-n (R) and L-n (R).

The use of drones or Unmanned Aerial Vehicles (UAVs) for aerial photography and cinematography is becoming widespread. The following optimization problem has recently been considered. Imagine a sporting event where a group of runners is competing and a team of drones with cameras is used to cover the event. The media director selects a set of filming scenes (determined by locations and time intervals), and the goal is to maximize the total filming time (the sum of recordings) achieved by the aerial cinematographers. Recently, it has been shown that this problem can be solved in polynomial time assuming the drones have unlimited battery endurance. In this paper, we prove that the problem is NP-hard for the more realistic case in which the battery endurance of the drones is limited.

Let G be a connected graph on n vertices and $$1 \le k \le n-1$$ 1 ≤ k ≤ n - 1 an integer. The k -token graph of G is the graph $$F_k(G)$$ F k ( G ) , whose vertices are all the k -subsets of vertices of G , two of which are adjacent whenever their symmetric difference is an edge of G . Every automorphism of G induces an automorphism of $$F_k(G)$$ F k ( G ) in a natural way. Suppose that $$S:=\{x,y\}$$ S : = { x , y } is a cut set of G , such that x and y have the same neighbours in $$G\setminus \{x,y\}$$ G \ { x , y } . In this paper, we show that there exists a large number of automorphisms of $$F_k(G)$$ F k ( G ) defined by S that are not induced by automorphisms of G . We also describe the group produced by all such 2-cuts of G .

A \emph{geometric graph} is a graph whose vertex set is a set of points in general position in the plane, and its edges are straight line segments joining these points. We show that for every integer $k \ge 2$, there exists a constat $c>0$ such that the following holds. The edges of every dense geometric graph can be colored with $k$ colors, such that the number of pairs of edges of the same color that cross is at most $(1/k-c)$ times the total number of pairs of edges that cross. The case when $k=2$ and $G$ is a complete geometric graph, was proved by Aichholzer et al.[\emph{GD} 2019].

Let G be a graph on n vertices and S a subset of vertices of G; the boundary of S is the set, ∂ S, of edges of G connecting S to its complement in G. The isoperimetric number of G, is the minimum of | ∂ S |/| S | overall S ⊂ V(G) of at most n/2 vertices. Let k ≤ n be positive integers. The Johnson graph is the graph, J(n,k), whose vertices are all the subsets of size k of {1,…,n}, two of which are adjacent if their intersection has cardinality equal to k-1. In this paper we show that the asymptotic value of the isoperimetric number of the Johnson graph J(n,2) is equal to (2-√(2))n.

Let $G$ be a graph of order $n$ and let $k\in \{1,2,\ldots,n-1\}$. The $k$-token graph of $G$ is the graph, whose vertices are all the $k$-subsets of vertices of $G$, where two such $k$-sets are adjacent whenever their symmetric difference is an edge of $G$. In this paper, we determine the automorphism group of the $k$-token graph of the complete bipartite graph $K_{m,n}$.

The k-token graph of G is the graph, F_k(G), whose vertices are all the k-subsets of V(G); with two of them adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this paper, we study the domination number of the token graphs of the star, S_n, and the complete graph, K_n.

The ability to avoid collisions with moving robots is critical in many applications. Moreover, if the robots have limited battery life, the goal is not only to avoid collisions but also to design efficient trajectories in terms of energy consumption and total mission time. This paper proposes a novel strategy for assigning turn angles for collision-free path planning in scenarios where a small team of robots cooperate in a certain mission. The algorithm allows each robot to reach a predetermined destination safely. It establishes consecutive, short time intervals, and at each interval, possible conflicts are solved centrally in an optimal manner. This is done by keeping constant speeds but generating a discrete set of possible directions for each robot, and solving efficiently the turn-angle allocation for a collision-free path that minimizes the path deviation from the shortest one. Due to the discretization, the final paths are not optimal, but the system can react to possible failures during execution, as conflicts are resolved at each time interval. Computational results and Software-In-The-Loop simulations are presented in order to evaluate the proposed algorithm. A comparison with a state-of-the-art approach shows that our algorithm is more energy-efficient and achieves lower mission completion time.

Let $S$ be a set of four points chosen independently, uniformly at random from a square. Join every pair of points of $S$ with a straight line segment. Color these edges red if they have positive slope and blue, otherwise. We show that the probability that $S$ defines a pair of crossing edges of the same color is equal to $1/4$. This is connected to a recent result of Aichholzer et al. [GD 2019] who showed that by 2-colouring the edges of a geometric graph and counting monochromatic crossings instead of crossings, the number of crossings can be more than halfed. Our result shows that for the described random drawings, there is a coloring of the edges such that the number of monochromatic crossings is in expectation $\frac{1}{2}-\frac{7}{50}$ of the total number of crossings.

Let $S$ be a set of $n$ points in general position in the plane. The Second Selection Lemma states that for any family of $\Theta(n^3)$ triangles spanned by $S$, there exists a point of the plane that lies in a constant fraction of them. For families of $\Theta(n^{3-\alpha})$ triangles, with $0\le \alpha \le 1$, there might not be a point in more than $\Theta(n^{3-2\alpha})$ of those triangles. An empty triangle of $S$ is a triangle spanned by $S$ not containing any point of $S$ in its interior. B\'ar\'any conjectured that there exist an edge spanned by $S$ that is incident to a super constant number of empty triangles of $S$. The number of empty triangles of $S$ might be $O(n^2)$; in such a case, on average, every edge spanned by $S$ is incident to a constant number of empty triangles. The conjecture of B\'ar\'any suggests that for the class of empty triangles the above upper bound might not hold. In this paper we show that, somewhat surprisingly, the above upper bound does in fact hold for empty triangles. Specifically, we show that for any integer $n$ and real number $0\leq \alpha \leq 1$ there exists a point set of size $n$ with $\Theta(n^{3-\alpha})$ empty triangles such that any point of the plane is only in $O(n^{3-2\alpha})$ empty triangles.

For sets of n points, n even, in general position in the plane, we consider straight-line drawings of perfect matchings on them. It is well known that such sets admit at least C_n/2 different plane perfect matchings, where C_n/2 is the n /2-th Catalan number. Generalizing this result we are interested in the number of drawings of perfect matchings which have k crossings. We show the following results. (1) For every k≤1/64n^2-35/32n√(n)+1225/64n , any set with n points, n sufficiently large, admits a perfect matching with exactly k crossings. (2) There exist sets of n points where every perfect matching has at most 5/72n^2-n/4 crossings. (3) The number of perfect matchings with at most k crossings is superexponential in n if k is superlinear in n . (4) Point sets in convex position minimize the number of perfect matchings with at most k crossings for k=0,1,2 , and maximize the number of perfect matchings with ( [ n/2; 2 ]) crossings and with ( [ n/2; 2 ]) -1 crossings.

For sets of $$n = 2m$$ points in general position in the plane we consider straight-line drawings of perfect matchings on them. It is well known that such sets admit at least $$C_m$$ different plane perfect matchings, where $$C_m$$ is the m-th Catalan number. Generalizing this result we are interested in the number of drawings of perfect matchings which have k crossings. We show the following results. (1) For every $$k\le \frac{1}{64}n^2-O(n \sqrt{n})$$ , any set of n points, n sufficiently large, admits a perfect matching with exactly k crossings. (2) There exist sets of n points where every perfect matching has fewer than $$\frac{5}{72}n^2$$ crossings. (3) The number of perfect matchings with at most k crossings is superexponential in n if k is superlinear in n. (4) Point sets in convex position minimize the number of perfect matchings with at most k crossings for $$k=0,1,2$$ , and maximize the number of perfect matchings with $$\left( {\begin{array}{c}n/2\\ 2\end{array}}\right) $$ crossings and with $${\left( {\begin{array}{c}n/2\\ 2\end{array}}\right) }\!-\!1$$ crossings.

A group of cooperative aerial robots can be deployed to efficiently patrol a terrain, in which each robot flies around an assigned area and shares information with the neighbors periodically in order to protect or supervise it. To ensure robustness, previous works on these synchronized systems propose sending a robot to the neighboring area in case it detects a failure. In order to deal with unpredictability and to improve on the efficiency in the deterministic patrolling scheme, this paper proposes random strategies to cover the areas distributed among the agents. First, a theoretical study of the stochastic process is addressed in this paper for two metrics: the idle time, the expected time between two consecutive observations of any point of the terrain and the isolation time, the expected time that a robot is without communication with any other robot. After that, the random strategies are experimentally compared with the deterministic strategy adding another metric: the broadcast time, the expected time elapsed from the moment a robot emits a message until it is received by all the other robots of the team. The simulations show that theoretical results are in good agreement with the simulations and the random strategies outperform the behavior obtained with the deterministic protocol proposed in the literature.

A diamond is the graph that is obtained from removing an edge from the complete graph on $4$ vertices. A ($C_4$,diamond)-free graph is a graph that does not contain a diamond or a cycle on four vertices as induced subgraphs. Let $G$ be a connected ($C_4$,diamond)-free graph on $n$ vertices. Let $1 \le k \le n-1$ be an integer. The $k$-token graph, $F_k(G)$, of $G$ is the graph whose vertices are all the sets of $k$ vertices of $G$; two of which are adjacent if their symmetric difference is a pair of adjacent vertices in $G$. Let $F$ be a graph isomorphic to $F_k(G)$. In this paper we show that given only $F$, we can construct in polynomial time a graph isomorphic to $G$. Let $\operatorname{Aut}(G)$ be the automorphism group of $G$. We also show that if $k\neq n/2$, then $\operatorname{Aut}(G) \simeq \operatorname{Aut}(F_k(G))$; and if $k = n/2$, then $\operatorname{Aut}(G) \simeq \operatorname{Aut}(F_k(G)) \times \mathbb{Z}_2$.

A geometric graph is a graph whose vertices are points in general position in the plane and its edges are straight line segments joining these points. In this paper we give an $O(n^2 \log n)$ algorithm to compute the number of pairs of edges that cross in a geometric graph on $n$ points. For layered, and convex geometric graphs the algorithm takes $O(n^2)$ time.

We study the Laplacian spectrum of token graphs, also called symmetric powers of graphs. The k-token graph \(F_k(G)\) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this work, we give a relationship between the Laplacian spectra of any two token graphs of a given graph. In particular, we show that, for any integers h and k such that \(1\le h\le k\le \frac{n}{2}\), the Laplacian spectrum of \(F_h(G)\) is contained in the Laplacian spectrum of \(F_k(G)\). Besides, we obtain a relationship between the spectra of the k-token graph of G and the k-token graph of its complement \(\overline{G}\). This generalizes a well-known property for Laplacian eigenvalues of graphs to token graphs.

We study the Laplacian spectrum of token graphs, also called symmetric powers of graphs. The k-token graph Fk(G) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this paper, we give a relationship between the Laplacian spectra of any two token graphs of a given graph. In particular, we show that, for any integers h and k such that 1≤h≤k≤n2, the Laplacian spectrum of Fh(G) is contained in the Laplacian spectrum of Fk(G). We also show that the doubled odd graphs and doubled Johnson graphs can be obtained as token graphs of the complete graph Kn and the star Sn=K1,n−1, respectively. Besides, we obtain a relationship between the spectra of the k-token graph of G and the k-token graph of its complement G‾. This generalizes to tokens graphs a well-known property stating that the Laplacian eigenvalues of G are closely related to the Laplacian eigenvalues of G‾. Finally, the doubled odd graphs and doubled Johnson graphs provide two infinite families, together with some others, in which the algebraic connectivities of the original graph and its token graph coincide. Moreover, we conjecture that this is the case for any graph G and its token graph.