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We study the problem of exploration by a mobile entity (agent) of a class of highly dynamic networks, namely the carrier graphs (the C-graphs, modeling public transportation systems, among others). These are defined by a set of carriers following infinitely their prescribed route along the stations of the network. Flocchini, Mans, and Santoro [9] studied this problem in the case when the agent must always travel on the carriers and thus cannot wait on a station. They described the necessary and sufficient conditions for the problem to be solvable and proved that the optimal worst-case number of time units (and thus of moves) to explore a n -node C-graph of k carriers and maximal period p is in Θ (k p 2) in the general case. In this paper, we study the impact of the ability to wait at the stations. We exhibit the necessary and sufficient conditions for the problem to be solvable in this context, and we prove that waiting at the stations allows the agent to reduce the optimal worst-case number of moves by a multiplicative factor of at least Θ (p) , while the worst-case time complexity is reduced to Θ (n p). (In any connected carrier graph, we have n ≤ k p.) We also show some complementary optimal results in specific cases (same period for all carriers, highly connected C-graphs). Finally this new ability allows the agent to completely map the C-graph, in addition to just exploring it. [ABSTRACT FROM AUTHOR]

Abstract: A mobile agent (robot), modeled as a finite automaton, has to visit all nodes of a regular graph. How does the memory size of the agent (the number of states of the automaton) influence its exploration capability? In particular, does every increase of the memory size enable an agent to explore more graphs? We give a partial answer to this problem by showing that a strict gain of the exploration power can be obtained by a polynomial increase of the number of states. We also show that, for automata with few states, the increase of memory by even one state results in the capability of exploring more graphs. [Copyright &y& Elsevier]

A group of mobile robots (beachcombers) have to search collectively every point of a given domain. At any given moment, each robot can be in walking mode or in searching mode. It is assumed that each robot's maximum allowed searching speed is strictly smaller than its maximum allowed walking speed. A point of the domain is searched if at least one of the robots visits it in searching mode. The Beachcombers' Problem consists in developing efficient schedules (algorithms) for the robots which collectively search all the points of the given domain as fast as possible. We consider searching schedules in the following one-dimensional geometric domains: the cycle of a known circumference L , the finite straight line segment of a known length L , and the semi-infinite line [ 0 , + ∞). We first consider the online Beachcombers' Problem (i.e. the scenario when the robots do not know in advance the length of the segment to be searched), where the robots are initially collocated at the origin of a semi-infinite line. It is sought to design a schedule A with maximum speed S , defined as S = inf ℓ ⁡ ℓ t A (ℓ) , where t A (ℓ) denotes the time when the search of the segment [ 0 , ℓ ] is completed under A. We consider a discrete and a continuous version of the problem, depending on whether the infimum is taken over ℓ ∈ N ⁎ or ℓ ≥ 1. We prove that the LeapFrog algorithm, which was proposed in Czyzowicz et al. (2015) [12] , is in fact optimal in the discrete case. This settles in the affirmative a conjecture from that paper. We also show how to extend this result to the more general continuous online setting. For the offline version of the Beachcombers' Problem (i.e. the scenario when the robots know in advance the length of the segment to be searched), we consider the t-source Beachcombers' Problem (i.e. all robots start from a fixed number t ≥ 1 of starting positions) on the cycle and on the finite segment. For the t-source Beachcombers' Problem on the cycle, we show that the structure of the optimal solutions is identical to the structure of the optimal solutions to the 2 t -source Beachcombers' Problem on a finite segment. In consequence, by using results from Czyzowicz et al. (2014) [13] , we prove that the 1-source Beachcombers' Problem on the cycle is NP-hard, and we derive approximation algorithms for the problem. For the t-source variant of the Beachcombers' Problem on the cycle and on the finite segment, we also derive efficient approximation algorithms. One important contribution of our work is that, in all variants of the offline Beachcombers' Problem that we discuss, we allow the robots to change direction of movement and search points of the domain on both sides of their respective starting positions. This represents a significant generalization compared to the model considered in Czyzowicz et al. (2014) [13] , in which each robot had a fixed direction of movement that was specified as part of the solution to the problem. We manage to prove that changes of direction do not help the robots achieve optimality. [ABSTRACT FROM AUTHOR]

Abstract A pair of agents (robots) are moving in a graph with the goal of meeting at the same node or while traversing the same edge. An asynchronous adversary knows the prescribed walks of the two agents and is in complete control of the speed of each agent during its walk. We provide a complete characterization of pairs of walks that enforce rendezvous against an asynchronous adversary after traversing a given number of edges. The characterization is efficient in that it can be checked in polynomial time. We argue that the certificate of rendezvous enforcement that is produced by the checking algorithm contains a wealth of information on why rendezvous is enforced. [ABSTRACT FROM AUTHOR]