Parameterized Complexity of Simultaneous Planarity

TL;DR

This work analyzes the parameterized complexity of SEFE in the sunflower setting, proving that the problem is fixed-parameter tractable when parameterized by the number of input graphs $k$ together with either the vertex cover number $\mathrm{vc}(G^{\cup})$ or the feedback edge set number $\mathrm{fes}(G^{\cup})$ of the union graph $G^{\cup}$. It also shows para-NP-hardness for the twin cover number of $G^{\cup}$ and establishes NP-completeness for SEFE even when the shared graph is a tree with maximum degree $4$, underscoring strong intractability for several natural parameters of $G$. On the shared graph side, the paper provides targeted FPT results for combinations such as $\mathrm{vc}(G) + \Delta_{1}$ and $\mathrm{cv}(G) + \Delta$, and develops a detailed SPQR-based embedding framework with preprocessing, block embeddings, nesting, and face-compatible decompositions to realize these results. Overall, the findings map a comprehensive tractability landscape: certain union-graph parameters yield efficient algorithms in tandem with $k$, while many natural shared-graph parameters remain hard, guiding future work on tighter parameterizations and broader fixed-embedding strategies.

Abstract

Given $k$ input graphs $G_1, \dots ,G_k$, where each pair $G_i$, $G_j$ with $i \neq j$ shares the same graph $G$, the problem Simultaneous Embedding With Fixed Edges (SEFE) asks whether there exists a planar drawing for each input graph such that all drawings coincide on $G$. While SEFE is still open for the case of two input graphs, the problem is NP-complete for $k \geq 3$ [Schaefer, JGAA 13]. In this work, we explore the parameterized complexity of SEFE. We show that SEFE is FPT with respect to $k$ plus the vertex cover number or the feedback edge set number of the the union graph $G^\cup = G_1 \cup \dots \cup G_k$. Regarding the shared graph $G$, we show that SEFE is NP-complete, even if $G$ is a tree with maximum degree 4. Together with a known NP-hardness reduction [Angelini et al., TCS 15], this allows us to conclude that several parameters of $G$, including the maximum degree, the maximum number of degree-1 neighbors, the vertex cover number, and the number of cutvertices are intractable. We also settle the tractability of all pairs of these parameters. We give FPT algorithms for the vertex cover number plus either of the first two parameters and for the number of cutvertices plus the maximum degree, whereas we prove all remaining combinations to be intractable.

Figures (11)

• Figure 1: (a) Three planar graphs $G^{}$, $G^{}$, and $G^{}$ with the shared graph $G$ highlighted in black. (b) Planar drawings of the same three graphs, but the subgraph $G$ is drawn the same way in all three drawings.
• Figure 2: Complexity of SEFE parameterized by combinations of the parameters of \ref{['sec:sharedParameters']}, assuming that the number of input graphs $k$ is part of the input. Check marks indicate tractability and crosses indicate para-NP-hardness. If $k$ is fixed, then \ref{['thm:unionmaxdeg4']} does not imply hardness, as the reduction requires an unbounded number of input graphs; the corresponding cells of the table are therefore still open. All other results also hold in this case.
• Figure 3: (a) Vertices $u$ and $v$ of different types having different neighborhoods in the two input graphs. $G^{}$ is shown with thin black edges, $G^{}$ with thick blue edges, overlapping edges belong to $G$. (b) Vertices $u$ and $v$ of the same type.
• Figure 4: An example illustrating the proof of \ref{['rr:vc-union']} for $k = 2$ input graphs with vertices $u, v$ belonging to the same type of $\mathcal{P}_{\leq2}$. Vertex $v$ can be placed directly next to vertex $u$ in the face $f$ that $u$ is contained in without introducing any crossings in the input graphs.
• Figure 5: Contraction of a shared (a) or a -exclusive (b) edge $e$ in a $2$-chain. Because the chain contains an additional -exclusive edge $e'$ in (b), the endpoints $x$ and $y$ of $e$ are contained in different connected components of all other input graphs. The -exclusive edge $g$ can therefore be rerouted as illustrated.

Theorems & Definitions (22)

• Lemma 1
• Theorem 1
• Definition 2
• Lemma 3
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5: Angelini15
• Theorem 6
• Theorem 7