In this paper we study cycles in multiset permutations and parking functions. As combinatorial objects, multiset permutations are essential building blocks for mappings and permutations, while parking functions lie between mappings and permutations. We take both algebraic and analytic views in our investigation and present exact as well as asymptotic results. We point to a surprising correspondence between two statistics on multiset permutations, terminal closers and cyclic points, shedding light on the combinatorial structure.

We study a two-player game played on undirected graphs called Trail Trap , which is a variant of a game known as Partizan Edge Geography . One player starts by choosing any edge and moving a token from one endpoint to the other; the other player then chooses a different edge and does the same. Alternating turns, each player moves their token along an unused edge from its current vertex to an adjacent vertex, until one player cannot move and loses. We present an algorithm to determine which player has a winning strategy when the graph is a tree and partially characterize the trees on which a given player wins. Additionally, we show that it is NP-hard to determine if Player 2 has a winning strategy on Trail Trap from the starting position, even for connected bipartite planar graphs with maximum degree 4. We determine which player has a winning strategy for certain subclasses of complete bipartite graphs and grid graphs, and we propose several open problems for further study.

The structure of a mobile ad hoc network changes dynamically based on node positioning. We consider a setting in which nodes can communicate if they are within a prescribed distance of one another, giving rise to a communication network. An example is a swarm of small satellites that cooperate to perform tasks; such swarms are likely to become commonplace in space missions. In this paper, we consider the problem of adding a new node or repositioning a current node in the network while optimizing a given network parameter such as network reliability. Although there are infinitely many locations to place the new node in space, there are only finitely many possible changes to the communication network. We provide an algorithm that enumerates all possible network changes in time O($n$2log n) or O($n$3log n), for networks in 2- or 3-dimensional Euclidean space, respectively. We apply the proposed algorithm to a satellite swarm formation planning problem, where the goal is to maximize network reliability.