Multi-Parameter Analysis of Finding Minors and Subgraphs in Edge Periodic Temporal Graphs

TL;DR

This paper investigates the computational complexity of structural properties in edge periodic temporal graphs, focusing on problems like shortest traversal and minor/subgraph detection, providing parameterized algorithms and complexity analysis.

Contribution

It introduces a detailed complexity analysis and parameterized algorithms for fundamental problems in edge periodic temporal graphs, a compact representation of dynamic networks.

Findings

Complexity results for shortest traversal in EPGs

Parameterized algorithms for minor and subgraph problems

Identification of tractable and intractable cases

Abstract

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 $s_e$ determines in which time steps the edge is present, namely $e$ is present in time step $t$ if and only if $s_e$ contains a $1$ at position $t \mod |s_e|$. 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 new concept of EPGs such as what is the shortest traversal time between two vertices; is there a time step in which the graph (1) is minor-free; (2) contains a minor; (3) is subgraph-free; (4) contains a 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 parameterized algorithms.