A Graph Minors Approach to Temporal Sequences

We develop a structural approach to simultaneous embeddability in temporal sequences of graphs, inspired by graph minor theory. Our main result is a classification theorem for 2-connected temporal sequences: we identify five obstruction classes and show that every 2-connected temporal sequence is either simultaneously embeddable or admits a sequence of improvements leading to an obstruction. This structural insight leads to a polynomial-time algorithm for deciding the simultaneous embeddability of 2-connected temporal sequences.

The restriction to 2-connected sequences is necessary, as the problem is NP-hard for connected graphs, while trivial for 3-connected graphs. As a consequence, our framework also resolves the rooted-tree SEFE problem, a natural extension of the well-studied Sunflower SEFE.

Our results uncover a rich structural theory of temporal planarity, laying the groundwork for a temporal graph minors theory.