Cops and Robber (C&R)refers to the well known class of pursuit-evasion games played by two adversarial teams, the cops and the robber, on a static graph G in discrete rounds. We consider the case when, instead, the edge-set of the graph is allowed to change in time, possibly at every round. Specifically, the game is played on an infinite periodic sequence G = (G(0), ... , G(p-1))* of graphs on the same set V of n vertices: in round t, the topology of G is G(i )= (V, E-i) where i equivalent to t (mod p). As in the (extensively investigated) case of static graphs, the main concern is on the characterization of the class of periodic temporal graphs where k cops can capture the robber. Another important concern is how to determine whether or not a given number k of cops is sufficient for them to win in a given G, and the complexity of such a determination. Concentrating on the case of a single cop, we provide a characterization of copwin periodic temporal graphs, establishing several basic properties on their nature. We do so by using a compact representation of periodic temporal graphs, introducing the novel notion of augmented arenas, and using these structures to extend to the temporal domain classical C&R concepts such as covers and corners. Based on these results, we design an algorithm for determining if a periodic temporal graph is copwin in time O(p n2 +n m), where m = & sum;i is an element of Zp |Ei|. Observe that, when p = 1 (i.e., in the static case), the complexity becomes O(n m). We then consider the case of k > 1 cops. By shifting from a representation in terms of directed graphs to one in terms of directed multi-hypergraphs, we prove that all the fundamental properties of augmented arenas continue to hold, providing a characterization of k-copwin periodic graphs, as well as a general strategy to determine if a periodic graph is k-copwin. All our results are general, in the sense that they do not rely on any assumption on properties such as connectivity, symmetry, reflexivity held (or not held) by the individual graphs in the sequence. All our results are established for a unified version of the game that includes the standard games studied in the literature, both for undirected and directed graphs, and both when the players are fully active (i.e., they always move) and when they are not. They hold also for a variety of settings not considered in the literature. (c) 2025 Published by Elsevier B.V.

The Dancing problem requires a swarm of n autonomous mobile robots to form a sequence of patterns, i.e., perform a choreography. Existing work has proven that some crucial restrictions on choreographies and initial configurations (e.g., on repetitions of patterns, periodicity, symmetries, contractions/expansions) must hold so that the Dancing problem can be solved under certain robot models. Here, we prove that these necessary constraints can be dropped by considering the ℒ𝒰ℳℐ model (i.e., where robots are endowed with a light whose color can be chosen from a constant-size palette) under the quite unexplored sequential scheduler. We formalize the class of Universal Dancing problems which require a swarm of n robots starting from any initial configuration to perform a (periodic or finite) sequence of arbitrary patterns, only provided that each pattern consists of n vertices (including multiplicities). However, we prove that, to be solvable under ℒ𝒰ℳℐ , the length of the feasible choreographies is bounded by the compositions of n into the number of colors available to the robots. We provide an algorithm solving Universal Dancing by exploiting the peculiar capability of sequential robots to implement a distributed counter. Even assuming non-rigid movements, our algorithm ensures spatial homogeneity of the performed choreography.

A periodic graph 𝒢=(G_0, G_1, G_2, … ) with period p is an infinite periodic sequence of graphs G_i = G_i + p = (V,E_i) , where i ≥ 0 . The graph G=(V,∪ _i E_i) is called the footprint of 𝒢 . Recently, the arena where the Cops and Robber game is played has been extended from a graph to a periodic graph; in this case, the cop number is also the minimum number of cops sufficient for capturing the robber. We study the connections and distinctions between the cop number c(𝒢) of a periodic graph 𝒢 and the cop number c(G) of its footprint G and establish several facts. For instance, we show that the smallest periodic graph with c(𝒢) = 3 has at most 8 nodes; in contrast, the smallest graph G with c(G) = 3 has 10 nodes. We push this investigation by generating multiple examples showing how the cop numbers of a periodic graph 𝒢 , the subgraphs G_i and its footprint G can be loosely tied. Based on these results, we derive upper bounds on the cop number of a periodic graph from properties of its footprint such as its treewidth.

We study the computational power that oblivious robots operating in the plane have under sequential schedulers. We show that this power is much stronger than the obvious capacity these schedulers offer of breaking symmetry, and thus to create a leader. More precisely, we consider the class of pattern formation problems, and focus on the most general problem in this class, Universal Pattern Formation (UPF), which requires the robots to form any pattern given in input, starting from any initial configurations (where robots may occupy the same point). We first show that UPF is unsolvable under ℱ𝒮𝒴𝒩𝒞 , even if the robots are endowed with additional strong capabilities (multiplicity detection, rigid movement, agreement on coordinate systems, presence of a unique leader). On the other hand, we prove that, except for point formation (Gathering), UPF is solvable under any sequential scheduler without any additional assumptions. We then turn our attention to the Gathering problem, and prove that weak multiplicity detection is necessary and sufficient for solvability under sequential schedulers. The obtained results show that the computational power of the robots under ℱ𝒮𝒴𝒩𝒞 (where Gathering is solvable without any multiplicity detection) and that under sequential schedulers are orthogonal.

The Dancing problem requires a swarm of n autonomous mobile robots to form a sequence of patterns, aka perform a choreography. Existing work has proven that some crucial restrictions on choreographies and initial configurations (e.g., on repetitions of patterns, periodicity, symmetries, contractions/expansions) must hold so that the Dancing problem can be solved under certain robot models. Here, we prove that these necessary constraints can be dropped by considering the LUMI model (i.e., where robots are endowed with a light whose color can be chosen from a constant-size palette) under the quite unexplored sequential scheduler. We formalize the class of Universal Dancing problems which require a swarm of n robots starting from any initial configuration to perform a (periodic or finite) sequence of arbitrary patterns, only provided that each pattern consists of n vertices (including multiplicities). However, we prove that, to be solvable under LUMI, the length of the feasible choreographies is bounded by the compositions of n into the number of colors available to the robots. We provide an algorithm solving the Universal Dancing problem by exploiting the peculiar capability of sequential robots to implement a distributed counter mechanism. Even assuming non-rigid movements, our algorithm ensures spatial homogeneity of the performed choreography.

In networked systems supporting mobile agents, a particularly dangerous security threat facing the agents is the presence of a black hole (BH): a network host that destroys any incoming agent without leaving any trace. The problem, called BLACK HOLE SEARCH ( BHS), of efficiently determining the location of such a dangerous host has been extensively studied under a variety of different assumptions. In spite of their differences, the existing results share the same assumption that all the searching agents are reliable. In this paper, we start the investigation of the BHS problem when some of the searching agents are faulty in a malicious way. More precisely, we consider that up to f of the k searching agents are Byzantine: they may behave in an arbitrary manner, actively misleading other agents; furthermore, they are in collusion with the black hole, and immune to its destructive power. We study under what conditions the BHS problem can be solved in a synchronous network of arbitrary topology in spite of the malicious agents, examining the impact on complexity of two factors: the a-priori topological knowledge held by the agents, and the communication mechanism available to them. We prove that, with prior knowledge about the graph topology (i.e., a network map), BHS can be solved by k >= 2f + 2 agents in O(n + f) synchronous rounds both with whiteboards and with just local communication, where n is the number of nodes in the network. Without any knowledge about the topological structure, using whiteboard communication BHS can be solved by k >= (f+1)(Delta+ 1) agents in O(m+f) rounds; instead, using local communication, BHS can be solved by k >= (f + 1)(Delta + 1) + 3f + 1 agents in O(m center dot n + f) rounds, where m is the number of links of the network and Delta is the maximum degree of the network. In all cases, as we show, the bound on the total number k of agents is asymptotically optimal.

Distributed computing is the science of cooperation. More precisely a distributed algorithm is composed of a set of computing entities (imposed to the programmers) each providing its own input, that have to cooperate to a common goal depending on the set of inputs (usually in the presence of adversaries such as asynchrony and failures) [2, 3]. . If follows that, in distributed computing, possibility/impossibility results and algorithmic techniques differ from what is encountered in sequential or parallel computing. This characterizes the distributed computing community. DISC and PODC are the very top conferences devoted to distributed computing.

We consider distributed systems of autonomous, punctiform, mobile robots that operate in the Euclidean plane by executing an infinite sequence of Look-Compute-Move cycles. Robots are anonymous, indistinguishable, homogeneous, and disoriented. In literature, four base models have been proposed to study four different memory-communication settings: 𝒪ℬℒ𝒪𝒯 (oblivious and silent), ℱ𝒮𝒯𝒜 (finite-state and silent), ℱ𝒞𝒪ℳ (oblivious and finite-communication), and ℒ𝒰ℳℐ (finite-state and finite-communication). In particular, the research has investigated how the computational power of these models is affected by considering three main classes of robot schedulers: FSYNCH (fully synchronous), SSYNCH (semi-synchronous), and ASYNCH (asynchronous). This paper focuses on a peculiar type of SSYNCH schedulers, the sequential ones, which activate only one robot at each round. We consider three subclasses: the general sequential scheduler (SEQ), the permutation scheduler (PERM), and the well-known round-robin (RROBIN). For each base model, we investigate how the robots’ computational power changes as the scheduler class varies, thus providing a first overview of the computational landscape of sequential schedulers.

In networked environments supporting mobile agents, a pressing problem is the presence of network sites harmful for the agents. In this paper we consider the danger posed by a node that destroys any incoming agent without leaving any trace. Such a dangerous node is known in the literature as a black hole (Bh). The problem of a team of system agents determining its location, known as black hole search (Bhs ), has been extensively studied in the literature under a variety of assumptions, both in synchronous and asynchronous settings. The main complexity parameter of Bhsis the number of system agents (called size) needed to solve the problem; other parameters are the number of moves (called cost) performed by the agents, and the time until termination.In the existing literature, with only a couple of exceptions, all results are based on a common assumption that the network is static, i.e. its topology does not change in time. We consider instead the Bhswhen the network is dynamic: the link structure of the graph changes over time. While time-varying graphs have been the focus of intense research in the last two decades, very little is known on the problem of locating the Bh in such networks.In this paper, we contribute to fill this research gap by studying Bhsin dynamic ring networks, focusing on the 1-interval connectivity adversarial dynamics. Feasibility and complexity of the problem depend on many factors, specifically on the size n of the ring, whether or not n is known, and the type of inter-agent communication (whiteboards, tokens, face-to-face, visual). In this paper, we provide a complete feasibility characterization presenting size optimal algorithms. Furthermore, we establish lower bounds on the cost and time of size-optimal solutions and show that our algorithms achieve those bounds.

We consider distributed systems of autonomous robots operating in the plane under synchronous Look-Compute-Move (LCM) cycles. Prior research on four distinct models assumes robots have unlimited energy. We remove this assumption and investigate systems where robots have limited but renewable energy, requiring inactivity for energy restoration. We analyze the computational impact of this constraint, fully characterizing the relationship between energy-restricted and unrestricted robots. Surprisingly, we show that energy constraints can enhance computational power. Additionally, we study how memory persistence and communication capabilities influence computation under energy constraints. By comparing the four models in this setting, we establish a complete characterization of their computational relationships. A key insight is that energy-limited robots can be modeled as unlimited-energy robots controlled by an adversarial activation scheduler. This provides a novel equivalence framework for analyzing energy-constrained distributed systems. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Consider a team of mobile computational entities, called agents, operating in a networked environment. To cooperatively solve problems and perform assigned tasks, the agents need to coordinate and communicate. The interaction mechanisms available to the agents have a strong impact on both computability and complexity. In the literature, a wide range of communication mechanisms have been explored, ranging from powerful options that require significant system resources (such as whiteboards) to very basic and cost-effective methods, like identical passive objects known as tokens. While algorithms that use whiteboards can effectively and explicitly communicate complex and resource-intensive information (like network maps), protocols that rely on token manipulation avoid to perform complex communication. Instead, they employ coordination and limited communication in an ad-hoc and implicit manner. The contribution of this paper consists in the design of simple, efficient communication protocols employing only tokens (and space). These protocols can be used as subroutines for explicitly communicating complex information in systems providing only token-based communication. This allows designers of token-based algorithms to explicitly employ complex communication between agents, enhancing their design freedom and flexibility. Additionally, our protocols facilitate the translation of results from models that depend solely on face-to-face communication to those that operate without such interactions but use tokens instead.

We consider distributed computations, by identical autonomous mobile entities, that solve the Point Convergence problem: given an arbitrary initial configuration of entities, disposed in the Euclidean plane, move in such a way that, for all ε>0, a configuration is eventually reached and maintained in which the separation between all entities is at most ε. The problem has been previously studied in a variety of settings. Our study concerns the minimal assumptions under which entities, moving asynchronously with limited and unknown visibility range and subject to limited imprecision in measurements, can be guaranteed to converge in this way. We present an algorithm that solves Point Convergence, provided the degree of asynchrony is bounded by some arbitrarily large but fixed constant. This provides a strong positive answer to a decade old open question posed by Katreniak. We also prove that, in an otherwise comparable setting, Point Convergence is impossible with unbounded asynchrony. This serves to distinguish the power of bounded and unbounded asynchrony in the control of autonomous mobile entities, settling at the same time a long-standing question whether in the Euclidean plane synchronous entities are more powerful than asynchronous ones.

In this paper, we address the challenge of locating a black hole within a dynamic graph using a set of scattered agents, which start from arbitrary positions in the graph. A black hole is defined as a node that silently eliminates any agent that visits it, effectively modeling network failures such as a crashed host or a destructive virus. The black hole search problem is considered solved when at least one agent survives and possesses a complete map of the graph, including the precise location of the black hole. Our study focuses on the scenario where the underlying graph is a dynamic 1-interval connected ring: a ring graph where, in each round, one edge may be absent. Agents communicate with other agents using movable pebbles that can be placed on nodes. In this setting, we demonstrate that three agents are sufficient to identify the black hole in O(n^2) moves. Furthermore, we prove that this number of agents is optimal. Additionally, we establish that the complexity bound is tight, requiring Ω(n^2) moves for any algorithm solving the problem with three agents, even when stronger communication mechanisms, such as unlimited-size whiteboards on nodes, are available.

We consider the recently introduced model of autonomous computational mobile entities called unconscious colored robots. The entities are the traditional oblivious silent mobile robots operating in the Euclidean plane in Look-Compute-Move cycles. However, each robot has a permanent external mark (or color) from a finite set, visible by the other robots, but not by the robot itself. The basic problem for these robots is separation, requiring all the robots with the same color to separate from the other robots, each group forming a recognizable geometric shape (e.g., circle, point, line); this task must be performed in finite time, in spite of the robots being unconscious of their own color, unable to communicate, and oblivious. This problem has been studied and solved in the synchronous setting (SSS 2023). In this paper we show that the problem is solvable also under the more difficult asynchronous adversary, provided the robots agree on the orientation of one axis, and no robot is uniquely colored. The proof is constructive: we present a distributed algorithm that allows unconscious colored robots with one-axis agreement to separate into parallel lines under the asynchronous scheduler.

In this paper we investigate the problem of searching for a black hole in a dynamic graph by a set of scattered agents (i.e., the agents start from arbitrary locations of the graph). The black hole is a node that silently destroys any agent visiting it. This kind of malicious node nicely models network failures such as a crashed host or a virus that erases the visiting agents. The black hole search problem is solved when at least one agent survives, and it has the entire map of the graph with the location of the black hole. We consider the case in which the underlining graph is a dynamic 1-interval connected ring: a ring graph in which at each round at most one edge can be missing. We first show that the problem cannot be solved if the agents can only communicate by using a face-to-face mechanism: this holds for any set of agents of constant size, with respect to the size n of the ring. To circumvent this impossibility we consider agents equipped with movable pebbles that can be left on nodes as a form of communication with other agents. When pebbles are available, three agents can localize the black hole in O(n2) moves. We show that such a number of agents is optimal. We also show that the complexity is tight, that is O(n2) moves are required for any algorithm solving the problem with three agents, even with stronger communication mechanisms (e.g., a whiteboard on each node on which agents can write messages of unlimited size). To the best of our knowledge this is the first paper examining the problem of searching a black hole in a dynamic environment with scattered agents.

Consider a set of mobile computational elements, called robots, that are viewed as points, and operate in the Euclidean plane in synchronous rounds. The robots are oblivious (they forget all computations performed in previous rounds), silent (unable of direct communication), and anonymous (indistinguishable from the outside). Each robot is provided with a private coordinate system, and can determine the position of the other robots (performing a Look operation); it has an algorithm, which it executes (performing a Compute operation) to determine a destination point; and it can move towards the destination (performing a Move operation). The k-Grouping problem requires the robots, starting from an arbitrary initial configuration in the plane, to gather at k distinct locations, not chosen in advance, by performing Look-Compute-Move cycles, and no longer move. This simple problem is however unsolvable if all the robots execute the same algorithm. It has been recently shown that, were different subgroups of the robots to execute different algorithms, the problem remains still unsolvable if the number of the algorithms is less than k. In this paper we prove that this number is minimum: we design k distinct algorithms and prove that, if each is executed by an arbitrary non-empty subset of the robots, they will collectively be able to solve the problem; furthermore, they are able to do so under a weak assumption on the level of agreement among the local coordinate systems. We further prove that, without any agreement, the problem becomes unsolvable even with k + 1 different algorithms. However, if an unbounded number of algorithms are allowed such that each robot has a unique algorithm, then we can solve k-Grouping without any agreement.

In 1883, Lewis Carrol wrote a newspaper article to criticize how the second best player was determined in a tennis tournament, and to suggest how such a task could be done correctly. This article has been taken by Donald Knuth as the inspiration for efficiently determining the smallest t elements of a totally ordered set of size n using k-comparisons. In the ensuing research, optimal algorithms for some low values of k and t have been established, by Knuth and Aigner; for k = 2 and t < 3, a few new bounds have been established for special values of n. Surprisingly, very little else is known on this problem, in spite of its illustrious pedigree and its relationship to other classical problems (e.g., selection and sorting with k -sorters). Enticed by the undeniable beauty of the problem, and the obvious promise of fun, we have joined the investigative quest. The purpose of this paper is to share some new results obtained so far. We are glad to report advances in two directions.

Research on distributed computing by a team of identical mobile computational entities, called robots, operating in a Euclidean space in $\mathit{Look}$-$\mathit{Compute}$-$\mathit{Move}$ ($\mathit{LCM}$) cycles, has recently focused on better understanding how the computational power of robots depends on the interplay between their internal capabilities (i.e., persistent memory, communication), captured by the four standard computational models (OBLOT, LUMI, FSTA, and FCOM) and the conditions imposed by the external environment, controlling the activation of the robots and their synchronization of their activities, perceived and modeled as an adversarial scheduler. We consider a set of adversarial asynchronous schedulers ranging from the classical semi-synchronous (SSYNCH) and fully asynchronous (ASYNCH) settings, including schedulers (emerging when studying the atomicity of the combination of operations in the $\mathit{LCM}$ cycles) whose adversarial power is in between those two. We ask the question: what is the computational relationship between a model $M_1$ under adversarial scheduler $K_1$ ($M_1(K_1)$) and a model $M_2$ under scheduler $K_2$ ($M_2(K_2)$)? For example, are the robots in $M_1(K_1)$ more powerful (i.e., they can solve more problems) than those in $M_2(K_2)$? We answer all these questions by providing, through cross-model analysis, a complete characterization of the computational relationship between the power of the four models of robots under the considered asynchronous schedulers. In this process, we also provide qualified answers to several open questions, including the outstanding one on the proper dominance of SSYNCH over ASYNCH in the case of unrestricted visibility.

A \emph{periodic graph} ${\cal G}=(G_0, G_1, G_2, \dots)$ with period $p$ is an infinite periodic sequence of graphs $G_i = G_{i + p} = (V,E_i)$, where $i \geq 0$. The graph $G=(V,\cup_i E_i)$ is called the footprint of ${\cal G}$. Recently, the arena where the Cops and Robber game is played has been extended from a graph to a periodic graph; in this case, the \emph{cop number} is also the minimum number of cops sufficient for capturing the robber. We study the connections and distinctions between the cop number $c({\cal G})$ of a periodic graph ${\cal G}$ and the cop number $c(G)$ of its footprint $G$ and establish several facts. For instance, we show that the smallest periodic graph with $c({\cal G}) = 3$ has at most $8$ nodes; in contrast, the smallest graph $G$ with $c(G) = 3$ has $10$ nodes. We push this investigation by generating multiple examples showing how the cop numbers of a periodic graph ${\cal G}$, the subgraphs $G_i$ and its footprint $G$ can be loosely tied. Based on these results, we derive upper bounds on the cop number of a periodic graph from properties of its footprint such as its treewidth.