A Graph Minors Approach to Temporal Sequences

TL;DR

The paper develops a graph-minor–driven framework for simultaneous embeddability in temporal sequences, focusing on 2-connected graphs and encoding embedding constraints via embedding lists. It identifies five obstruction classes (disagreeable, amazingly-bad, gaudy, inconsistent, and uncombinable) and proves a structural theorem: every nonempty 2-connected temporal sequence with two lists is either obstructed or admits a sequence of improvements leading to an obstruction, enabling a polynomial-time decision algorithm. Core techniques include Tutte decompositions, stretching to nice accurate expansions, and a suite of analogue reductions (amazing/delightful) that preserve embeddability while simplifying instances. As a consequence, the rooted-tree SEFE problem is resolved in the 2-connected setting, and the work provides a versatile methodology for applying graph-minor concepts to evolving graphs, with open directions toward broader surfaces, multi-colour bond analyses, and algebraic extensions to matroids. Overall, the results establish a robust, scalable framework for temporal graph minor theory with practical algorithmic implications.

Abstract

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. More broadly, our results demonstrate the applicability of graph minor techniques to evolving graph structures and provide a foundation for future algorithmic and structural investigations in temporal graph theory.

Figures (14)

• Figure 1: A temporal triple $(G_1,G_2,G_3)$. We obtain $G_2$ from $G_1$ by contracting the edge $4$ and deleting $7$ and from $G_3$ by contracting the edge $8$ and deleting $9$. Depicted are two possible sequences of embeddings, one of which (top) is simultaneous whereas the other (bottom) fails to be consistent.
• Figure 2: A temporal triple $(G_1,G_2,G_3)$ which has no simultaneous embedding. The graph $G_2$ is obtained from $G_1$ by contracting the edge $5$ and is obtained from $G_3$ by contracting the edge $6$. Although $G_1$ and $G_3$ are isomorphic, they minor down to $G_2$ in different ways, as can be seen by considering the edges labelled $1,2,3$ and $4$.
• Figure 3: A temporal triple $(G_1,G_2,G_3)$ which has no simultaneous embedding. The graph $G_2$ is obtained from $G_1$ by deleting the edges $e$ and $f$ and is obtained from $G_3$ by deleting the edge $g$.
• Figure 4: Three temporal sequences ${\cal G}_1$, ${\cal G}_2$ and ${\cal G}_3$. The temporal sequence ${\cal G}_2$ is obtained from ${\cal G}_1$ by reducing the blue$2$-component at $\{x,y\}$ in $G_1$ to a single edge and 'uncontracting' some edges in $G_3$. The temporal sequence ${\cal G}_3$ is obtained from ${\cal G}_2$ by deleting the edge $e$ from $G_1$ and $G_2$ and 'uncontracting' some edges in $G_3$. Each temporal sequence has the same number of simultaneous embeddings.
• Figure 5: An illustration of the four ways in which a segment $S$ (in blue) can be affectively connected. The (cyclic or linear) structure suggested by $A$ is drawn as a red line.

Theorems & Definitions (139)

• Example 1
• Example 2
• Theorem 1.1
• Corollary 1
• Lemma 1: folklore graphsonsurfaces2001
• Lemma 2: Whitney 1933 whitney2iso
• Corollary 2
• Definition 1: trivial list
• Example 3
• Definition 2: $2$-component