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The mutual-visibility problem in a graph $G$ asks for the cardinality of a largest set of vertices $S\subseteq V(G)$ so that for any two vertices $x,y\in S$ there is a shortest $x,y$-path $P$ so that all internal vertices of $P$ are not in $S$. This is also said as $x,y$ are visible with respect to $S$, or $S$-visible for short. Variations of this problem are known, based on the extension of the visibility property of vertices that are in and/or outside $S$. Such variations are called total, outer and dual mutual-visibility problems. This work is focused on studying the corresponding four visibility parameters in graphs of diameter two, throughout showing bounds and/or closed formulae for these parameters. The mutual-visibility problem in the Cartesian product of two complete graphs is equivalent to (an instance of) the celebrated Zarankievicz's problem. Here we study the dual and outer mutual-visibility problem for the Cartesian product of two complete graphs and all the mutual-visibility problems for the direct product of such graphs as well. We also study all the mutual-visibility problems for the line graphs of complete and complete bipartite graphs. As a consequence of this study, we present several relationships between the mentioned problems and some instances of the classical Tur\'an problem. Moreover, we study the visibility problems for cographs and several non-trivial diameter-two graphs of minimum size., Comment: 23 pages, 4 figures

For a set of robots (or agents) moving in a graph, two properties are highly desirable: confidentiality (i.e., a message between two agents must not pass through any intermediate agent) and efficiency (i.e., messages are delivered through shortest paths). These properties can be obtained if the \textsc{Geodesic Mutual Visibility} (GMV, for short) problem is solved: oblivious robots move along the edges of the graph, without collisions, to occupy some vertices that guarantee they become pairwise geodesic mutually visible. This means there is a shortest path (i.e., a ``geodesic'') between each pair of robots along which no other robots reside. In this work, we optimally solve GMV on finite hexagonal grids $G_k$. This, in turn, requires first solving a graph combinatorial problem, i.e. determining the maximum number of mutually visible vertices in $G_k$., Comment: 24 pages, 13 figures

If $X$ is a subset of vertices of a graph $G$, then vertices $u$ and $v$ are $X$-visible if there exists a shortest $u,v$-path $P$ such that $V(P)\cap X \subseteq \{u,v\}$. If each two vertices from $X$ are $X$-visible, then $X$ is a mutual-visibility set. The mutual-visibility number of $G$ is the cardinality of a largest mutual-visibility set of $G$ and has been already investigated. In this paper a variety of mutual-visibility problems is introduced based on which natural pairs of vertices are required to be $X$-visible. This yields the total, the dual, and the outer mutual-visibility numbers. We first show that these graph invariants are related to each other and to the classical mutual-visibility number, and then we prove that the three newly introduced mutual-visibility problems are computationally difficult. According to this result, we compute or bound their values for several graphs classes that include for instance grid graphs and tori. We conclude the study by presenting some inter-comparison between the values of such parameters, which is based on the computations we made for some specific families., Comment: 23 pages, 4 figures, original paper

Let $G$ be a graph and $X\subseteq V(G)$. Then, vertices $x$ and $y$ of $G$ are $X$-visible if there exists a shortest $u,v$-path where no internal vertices belong to $X$. The set $X$ is a mutual-visibility set of $G$ if every two vertices of $X$ are $X$-visible, while $X$ is a total mutual-visibility set if any two vertices from $V(G)$ are $X$-visible. The cardinality of a largest mutual-visibility set (resp. total mutual-visibility set) is the mutual-visibility number (resp. total mutual-visibility number) $\mu(G)$ (resp. $\mu_t(G)$) of $G$. It is known that computing $\mu(G)$ is an NP-complete problem, as well as $\mu_t(G)$. In this paper, we study the (total) mutual-visibility in hypercube-like networks (namely, hypercubes, cube-connected cycles, and butterflies). Concerning computing $\mu(G)$, we provide approximation algorithms for both hypercubes and cube-connected cycles, while we give an exact formula for butterflies. Concerning computing $\mu_t(G)$ (in the literature, already studied in hypercubes), we provide exact formulae for both cube-connected cycles and butterflies., Comment: 19 pages, 8 figures

The \textsc{Mutual Visibility} is a well-known problem in the context of mobile robots. For a set of $n$ robots disposed in the Euclidean plane, it asks for moving the robots without collisions so as to achieve a placement ensuring that no three robots are collinear. For robots moving on graphs, we consider the \textsc{Geodesic Mutual Visibility} ($\GMV$) problem. Robots move along the edges of the graph, without collisions, so as to occupy some vertices that guarantee they become pairwise geodesic mutually visible. This means that there is a shortest path (i.e., a "geodesic") between each pair of robots along which no other robots reside. We study this problem in the context of finite and infinite square grids, for robots operating under the standard Look-Compute-Move model. In both scenarios, we provide resolution algorithms along with formal correctness proofs, highlighting the most relevant peculiarities arising within the different contexts, while optimizing the time complexity., Comment: 28 pages, 9 figures. To appear in the proceedings of the 25th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS 2023), Institute for Future Technologies, Jersey City, New Jersey, USA, October 2-4, 2023

The concept of mutual-visibility in graphs has been recently introduced. If $X$ is a subset of vertices of a graph $G$, then vertices $u$ and $v$ are $X$-visible if there exists a shortest $u,v$-path $P$ such that $V(P)\cap X \subseteq \{u, v\}$. If every two vertices from $X$ are $X$-visible, then $X$ is a mutual-visibility set. The mutual-visibility number of $G$ is the cardinality of a largest mutual-visibility set of $G$. It is known that computing the mutual-visibility number of a graph is NP-complete, whereas it has been shown that there are exact formulas for special graph classes like paths, cycles, blocks, cographs, and grids. In this paper, we study the mutual-visibility in distance-hereditary graphs and show that the mutual-visibility number can be computed in linear time for this class., Comment: 16 pages, 5 figures, a preliminary version will appear on the proc. of the XII Latin and American Algorithms, Graphs and Optimization Symposium, {LAGOS} 2023, Huatulco, Mexico, September 18-22, 2023. Procedia Computer Science, Elsevier

Given a set R of robots, each one located at different vertices of an infinite regular tessellation graph, we aim to explore the Arbitrary Pattern Formation (APF) problem. Given a multiset F of grid vertices such that |R|=|F|, APF asks for a distributed algorithm that moves robots so as to reach a configuration similar to F. Similarity means that robots must be disposed as F regardless of translations, rotations, reflections. So far, as possible graph discretizing the Euclidean plane only the standard square grid has been considered in the context of the classical Look-Compute-Move model. However, it is natural to consider also the other regular tessellation graphs, that are triangular and hexagonal grids. We provide a resolution algorithm for APF when the initial configuration is asymmetric and the considered topology is any regular tessellation graph.

Following the wide investigation in distributed computing issues by mobile entities of the last two decades, we consider the need of a structured methodology to tackle the arisen problems. The aim is to simplify both the design of the resolution algorithms and the writing of the required correctness proofs. We would encourage the usage of a common framework in order to help both algorithm designer and reviewers in the intricate work of analyzing the proposed resolution strategies. In order to better understand the potentials of our methodology, we consider the Pattern Formation (PF) problem approached in [Fujinaga et al. SIAM J. Comput., 2015] as case study. Since the proposed resolution algorithm has turned out to be inaccurate and also of difficult fixing, we design a new algorithm guided by the proposed methodology, hence fully characterizing the problem.

Given any multiset F of points in the Euclidean plane and a set R of robots such that |R|=|F|, the Arbitrary Pattern Formation (APF) problem asks for a distributed algorithm that moves robots so as to reach a configuration similar to F. Similarity means that robots must be disposed as F regardless of translations, rotations, reflections, uniform scalings. Initially, each robot occupies a distinct position. When active, a robot operates in standard LCM cycles. Robots are asynchronous, oblivious, anonymous, silent and execute the same distributed algorithm. So far, the problem has been mainly addressed by assuming chirality, that is robots share a common left-right orientation. We are interested in removing such a restriction. While working on the subject, we faced several issues that required close attention. We deeply investigated how such difficulties were overcome in the literature, revealing that crucial arguments for the correctness proof of the existing algorithms have been neglected. The systematic lack of rigorous arguments with respect to necessary conditions required for providing correctness proofs deeply affects the validity as well as the relevance of strategies proposed in the literature. Here we design a new deterministic distributed algorithm that fully characterizes APF showing its equivalence with the well-known Leader Election problem in the asynchronous model without chirality. Our approach is characterized by the use of logical predicates in order to formally describe our algorithm as well as its correctness. In addition to the relevance of our achievements, our techniques might help in revising previous results. In fact, it comes out that well-established results like [Fujinaga et al, SIAM J. Comp. 44(3) 2015], more recent approaches like [Bramas et al, PODC and SSS 2016] and 'unofficial' results like [Dieudonne et al, arXiv:0902.2851] revealed to be not correct., Comment: To appear in Distributed Computing

Recently, the recoverable robustness model has been introduced in the optimization area. This model allows to consider disruptions (input data changes) in a unified way, that is, during both the strategic planning phase and the operational phase. Although the model represents a significant improvement, it has the following drawback: we are typically not facing only one disruption, but many of them might appear one after another. In this case, the solutions provided in the context of the recoverable robustness are not satisfying. In this paper we extend the concept of recoverable robustness to deal not only with one single recovery step, but with arbitrarily many recovery steps. To this aim, we introduce the notion of dynamic recoverable robustness problems. We apply the new model in the context of timetabling and delay management problems. We are interested in finding efficient dynamic robust algorithms for solving the timetabling problem and in evaluating the price of robustness of the proposed solutions.

Recently, the recoverable robustness model has been introduced in the optimization area. This model allows to consider disruptions (input data changes) in a unified way, that is, during both the strategic planning phase and the operational phase. Although the model represents a significant improvement, it has the following drawback: we are typically not facing only one disruption, but many of them might appear one after another. In this case, the solutions provided in the context of the recoverable robustness are not satisfying. In this paper we extend the concept of recoverable robustness to deal not only with one single recovery step, but with arbitrarily many recovery steps. To this aim, we introduce the notion of dynamic recoverable robustness problems. We apply the new model in the context of timetabling and delay management problems. We are interested in finding efficient dynamic robust algorithms for solving the timetabling problem and in evaluating the price of robustness of the proposed solutions.

In this paper we provide efficient robust algorithms for shunting problems concerning the reordering of train cars over a hump. In particular, we study algorithms able to cope with small disruptions, as temporary and local availability and/or malfunctioning of key resources that can occur and affect planned operations. To this aim, a definition of robust algorithm is provided. Performances of the proposed algorithms are measured by the notion of price of robustness. Various scenarios are considered, and interesting results are presented.

In railways systems the timetable is typically represented as a weighted digraph on which itinerary queries are answered by shortest path algorithms, usually running Dijkstra's algorithm. Due to the continuously growing size of real-world graphs, there is a constant need for faster algorithms and many techniques have been devised to heuristically speed up Dijkstra's algorithm. One of these techniques is the multi-level overlay graph, that has been recently introduced and shown to be experimentally efficient, especially when applied to timetable information. In many practical application major disruptions to the normal operation cannot be completely avoided because of the complexity of the underlying systems. Timetable information update after disruptions is considered one of the weakest points in current railway systems, and this determines the need for an effective online redesign and update of the shortest paths information as a consequence of disruptions. In this paper, we make a step forward toward this direction by showing some theoretical properties of multi-level overlay graphs that lead us to the definition of a new data structure for the dynamic maintenance of a multi-level overlay graph of a given graph G while weight decrease or weight increase operations are performed on G. Our solution is theoretically faster than the recomputation from scratch and allows fast queries.

In this paper we provide efficient robust algorithms for shunting problems concerning the reordering of train cars over a hump. In particular, we study algorithms able to cope with small disruptions, as temporary and local availability and/or malfunctioning of key resources that can occur and affect planned operations. To this aim, a definition of robust algorithm is provided. Performances of the proposed algorithms are measured by the notion of price of robustness. Various scenarios are considered, and interesting results are presented.

In railways systems the timetable is typically represented as a weighted digraph on which itinerary queries are answered by shortest path algorithms, usually running Dijkstra's algorithm. Due to the continuously growing size of real-world graphs, there is a constant need for faster algorithms and many techniques have been devised to heuristically speed up Dijkstra's algorithm. One of these techniques is the multi-level overlay graph, that has been recently introduced and shown to be experimentally efficient, especially when applied to timetable information. In many practical application major disruptions to the normal operation cannot be completely avoided because of the complexity of the underlying systems. Timetable information update after disruptions is considered one of the weakest points in current railway systems, and this determines the need for an effective online redesign and update of the shortest paths information as a consequence of disruptions. In this paper, we make a step forward toward this direction by showing some theoretical properties of multi-level overlay graphs that lead us to the definition of a new data structure for the dynamic maintenance of a multi-level overlay graph of a given graph G while weight decrease or weight increase operations are performed on G. Our solution is theoretically faster than the recomputation from scratch and allows fast queries.