Weak Deletion Sequence Analysis

• Weak deletion sequences are defined as finite chains of simple graphs where each graph is isomorphic to a subgraph of its adjacent graph, capturing both insertion and deletion transitions.
• They model temporal graph dynamics by allowing a single operation type (insertion or deletion) per step, which is crucial for understanding transitions in temporal networks.
• The topic reveals that determining simultaneous planar embeddability for 2-connected graphs in weak deletion sequences is NP-hard, contrasting with polynomial-time solutions for strict deletion sequences.

A weak deletion sequence is a finite, time-ordered chain of simple graphs $(G_1, G_2, \dots, G_n)$ such that, at every step, either $G_i$ is isomorphic to a subgraph of $G_{i+1}$ or vice versa: $G_{i+1}$ is isomorphic to a subgraph of $G_i$. The primary significance of weak deletion sequences lies in the theory of temporal graphs, where they capture models in which the transition from one graph to the next is allowed to be either an insertion or a deletion (up to isomorphism) but prohibits both operations simultaneously within a single step. Determining the simultaneous planar embeddability of weak deletion sequences of 2-connected graphs is a computational problem recently shown to be NP-hard (Carmesin et al., 6 Dec 2025).

1. Formal Definition

A weak deletion sequence is defined as follows:

A sequence of simple graphs $(G_{1}, G_{2}, \dots, G_{n})$ is a weak deletion sequence if for every $i \in \{1, \ldots, n-1\}$, one of the following holds:

• $G_i$ is isomorphic to a subgraph of $G_{i+1}$,
• $G_{i+1}$ is isomorphic to a subgraph of $G_i$.

Equivalently, for each $i$, there exists an isomorphism $\phi$ and an inclusion of edge-sets such that one graph is a minor of the other by deletion of edges and isolated vertices, but without contracting edges or mixing deletion with contraction in the same step.

A sequence is said to be simultaneously embeddable if there exists a collection of plane-embeddings $\iota_{i}$ of each $G_i$ such that, whenever $G_i \cong H \subseteq G_{i+1}$, the embedding $\iota_{i+1}$ of $G_{i+1}$ induces—up to isomorphism—the embedding $\iota_i$ on $H$; symmetrically for $G_{i+1} \cong H' \subseteq G_{i}$.

2. Illustrative Example

Consider a three-step weak deletion sequence $(G_1, G_2, G_3)$ (with all graphs considered up to isomorphism):

• $G_1$: a triangle $u_1u_2u_3u_1$ plus a pendant edge $u_3v$,
• $G_2$: triangle $u_1u_2u_3$, obtained from $G_1$ by deleting $u_3v$,
• $G_3$: $G_2$ plus a leaf $w$ attached to both $u_1$ and $u_2$.

$G_2$ is isomorphic to a subgraph of both $G_1$ and $G_3$, so the sequence satisfies the weak deletion property. For simultaneous embeddability, the embedding of $G_2$ must coincide with the induced embedding from both $G_1$ and $G_3$. In this case, since a triangle has a unique planar embedding up to a face flip, and both $G_1$ and $G_3$ induce the same cyclic order on the triangle, the sequence is simultaneously embeddable.

3. Structural Lemmas and Hybrid Minor Analogue

The analysis of weak deletion sequences is facilitated by introducing a "hybrid" minor relation, which allows mixing strict (non-relabelable) edges with weak (isomorphic) edges.

• Hybrid Equivalence (Lemma 1.2): Any weak deletion sequence of 2-connected graphs can be transformed in polynomial time into a hybrid deletion sequence—by replacing each weak edge with a suitably large rigid gadget—so that simultaneous embeddability is preserved. Therefore, NP-hardness for hybrids implies NP-hardness for weak sequences (Carmesin et al., 6 Dec 2025).

The construction of complex Boolean relations over the embeddings across the sequence is achieved through planar gadgetry:

• The Equaliser gadget (length 9) enforces equality of inside/outside face allocations between pairs of "houses."
• The Negator gadget (length 9) enforces difference.
• The Or gadget (length 15, three houses) enforces that at least one house is inside.

By concatenating these gadgets and combining them in noncrossing ways around a common cycle (as ensured by Lemma 4.8), arbitrary 3-SAT formulas can be encoded, crucially supporting NP-hardness reductions.

4. Complexity Landscape

The complexity results concerning weak deletion sequences are as follows:

Case	Complexity	Notes
Weak deletion sequence (2-connected graphs)	NP-hard	By reduction from 3-SAT, even for length $\leq 95$
Strict deletion sequence (2-connected graphs)	Polynomial time	Solvable via dynamic programming on triconnected components
Weak/Strict deletion (connected graphs)	NP-hard	Both variants hard even for connected graphs
Weak/Strict deletion (3-connected graphs)	Trivial	Unique embedding; always embeddable

For 2-connected graphs, the weak variant forms the complexity boundary: allowing isomorphisms in succession elevates the problem from polynomial-time solvable (strict variant) to NP-hard (weak variant). For connected graphs, all variants are NP-hard, while for 3-connected planar graphs every pair of successive graphs has a unique planar embedding, making the sequence trivially simultaneously embeddable (Carmesin et al., 6 Dec 2025).

5. Gadget Frameworks and Encoding Boolean Constraints

The NP-hardness reduction is implemented via the construction of specific planar gadgets:

• Equaliser: Forces two houses to share the same face assignment.
• Negator: Forces two houses to differ.
• Or: Ensures at least one among three houses is "inside."

These gadgets can be combined by gluing along a common cycle, provided their "arch structures" do not cross. Noncrossing arrangements guarantee a unique way to assemble the embeddings (Lemma 4.8), thus enabling the encoding of complex constraint systems such as arbitrary 3-SAT instances. This construction yields NP-hardness even for bounded sequence length.

6. Further Remarks and Open Directions

The hybrid-minor framework permits modeling of mixed strict and weak relabeling operations and may hold broader applicability beyond planarity. In the extended setting of indefinite temporal sequences, where the type of minor operation (deletion or contraction) is unspecified at each step, NP-hardness persists. The strict-sequence theory for 2-connected graphs enables an efficient dynamic-programming algorithm via triconnected-component decomposition, offering a tractable positive result for this special case (Carmesin et al., 6 Dec 2025).

Open research avenues include:

• Parameterized complexity analysis with respect to tree-width of the minor-order or the sequence length,
• Dichotomy theorems for other classes of minor-like temporal graph models.

A plausible implication is that the hybrid minor framework and the combinatorics of gadget construction could find further applications in temporal graph theory and related domains.

7. Significance and Boundary of Tractability

By formalizing the notion of weak deletion sequences and developing gadgets whose allocation sets implement any desired Boolean relation, it is established that planarity testing for 2-connected weak deletion sequences marks the transition from tractable (polynomial-time) to intractable (NP-hard) in the landscape of temporal and minor-related planar embedding problems. This boundary underscores the algorithmic sensitivity of planarity constraints to the relaxation of minor operations—specifically, the allowance of isomorphisms in sequential graph morphisms (Carmesin et al., 6 Dec 2025).