Multi-parameter analysis of finding minors and induced subgraphs in edge-periodic temporal graphs

We study the computational complexity of determining structural properties of edge-periodic temporal graphs (EPGs). EPGs are time-varying graphs that compactly represent periodic behavior of components of a dynamic network, for example, train schedules on a rail network. In EPGs, for each edge e of the graph, a binary string τ(e) determines in which time steps the edge is present, namely e is present in time step t if and only if τ(e) contains a 1 at position tmod|τ(e)|. Due to this periodicity, EPGs serve as very compact representations of complex periodic systems and can even be exponentially smaller than classic temporal graphs representing one period of the same system, as the latter contain the whole sequence of graphs explicitly. In this paper, we study the computational complexity of fundamental questions of the concept of EPGs such as: Is there a time step or a sliding window of size Δ in which the graph (1) is minor-free; (2) contains a minor; (3) is induced subgraph-free; (4) contains an induced subgraph; with respect to a given minor or subgraph. We give a detailed parameterized analysis for multiple combinations of parameters for the problems stated above including several algorithms. Additionally, we study the parameterized complexity of the short traversal problem in EPGs. In this problem, one asks whether there exists a time step t such that one can reach a vertex b from a vertex a at time step at most t+k for given k.