Determining the Outerthickness of Graphs Is NP-Hard

We give a short, self-contained, and easily verifiable proof that determining the outerthickness of a general graph is NP-hard. This resolves a long-standing open problem on the computational complexity of outerthickness. Moreover, our hardness result applies to a more general covering problem $P_F$, defined as follows. Fix a proper graph class $F$ whose membership is decidable. Given an undirected simple graph $G$ and an integer $k$, the task is to cover the edge set $E(G)$ by at most $k$ subsets $E_1,\ldots,E_k$ such that each subgraph $(V(G),E_i)$ belongs to $F$. Note that if $F$ is monotone (in particular, when $F$ is the class of all outerplanar graphs), any such cover can be converted into an edge partition by deleting overlaps; hence, in this case, covering and partitioning are equivalent. Our result shows that for every proper graph class $F$ whose membership is decidable and that satisfies all of the following conditions: (a) $F$ is closed under topological minors, (b) $F$ is closed under $1$-sums, and (c) $F$ contains a cycle of length $3$, the problem $P_F$ is NP-hard for every fixed integer $k\ge 3$. In particular: For $F$ equal to the class of all outerplanar graphs, our result settles the long-standing open problem on the complexity of determining outerthickness. For $F$ equal to the class of all planar graphs, our result complements Mansfield’s NP-hardness result for the thickness, which applies only to the case $k=2$. It is also worth noting that each of the three conditions above is necessary. If $F$ is the class of all eulerian graphs, then cond. (a) fails. If $F$ is the class of all pseudoforests, then cond. (b) fails. If $F$ is the class of all forests, then cond. (c) fails. For each of these three classes $F$, the problem $P_F$ is solvable in polynomial time for every fixed integer $k\ge 3$, showing that none of the three conditions can be dropped.

💡 Research Summary

The paper settles a long‑standing open question in graph theory: determining the outerthickness of a general graph is NP‑hard for any fixed integer k ≥ 3. Outerthickness is defined as the smallest number of edge‑disjoint outerplanar subgraphs whose union equals the original graph. While the analogous thickness problem (planar subgraphs) was shown NP‑hard by Mansfield in 1983, the outerplanar restriction resisted classification for decades.

The authors first formalize the decision version, OuterThickness(G, k): given a simple undirected graph G and an integer k ≥ 1, does there exist a partition of E(G) into k parts each inducing an outerplanar graph? They then introduce a more general covering framework, denoted P_F, where a fixed graph class F (with decidable membership) replaces “outerplanar”. The task is to cover the edge set of G with at most k subsets so that each induced subgraph belongs to F. If F is monotone (e.g., outerplanar graphs) the covering can be turned into a partition, making the two notions equivalent.

The central theorem (Theorem 3) states that for any proper graph class F that satisfies three natural conditions—(a) closed under topological minors, (b) closed under 1‑sums, and (c) contains a triangle—P_F is NP‑hard for every fixed k ≥ 3. Moreover, if membership in F can be decided in polynomial time, then P_F is NP‑complete. By instantiating F as the class of outerplanar graphs, the authors obtain NP‑completeness of OuterThickness for k ≥ 3, finally resolving the open problem. By taking F as the class of planar graphs, they extend Mansfield’s thickness hardness from k = 2 to all k ≥ 3.

The hardness proof is built on a reduction from edge‑coloring of regular graphs, a classic NP‑complete problem. Specifically, for k = 3 they use Holyer’s result that 3‑edge‑coloring of 3‑regular graphs is NP‑complete; for general k ≥ 3 they use the Leven–Galil result for k‑regular graphs. Given a k‑regular input graph G, the reduction constructs a labeling function φ that assigns a distinct label to each edge in any path of length at most three. This labeling is guaranteed by a combinatorial construction that first builds a constant‑size “gadget” graph H which is edge‑maximal with outerthickness k and contains a large independent set. The vertices of this independent set serve as “attachment points” for the edges of G.

The final constructed graph G′ is obtained by taking disjoint copies of G and H, then for each edge e = {u,v} in G adding two new edges {u, w_{φ(e)}} and {v, w_{φ(e)}} where w_{φ(e)} is the independent‑set vertex indexed by the label of e. This operation preserves simplicity because the labeling guarantees that incident edges at any vertex receive distinct labels, preventing parallel edges.

A key technical observation (Observation 6) describes how adding a path and two “dangling” edges to an edge‑maximal outerplanar graph Q behaves: if the two dangling edges attach to the same vertex, the resulting graph remains outerplanar; if they attach to distinct non‑adjacent vertices, outerplanarity is destroyed. This observation underpins a series of claims (Claims 8–10) that force any outerplanar partition of G′ into k parts to respect the original edge‑coloring structure of G. In particular, each vertex of G must have its three incident new edges placed in three different parts, and each original edge together with its two incident new edges must all lie in the same part. Violating these constraints would create a subgraph of the form described in Observation 6(b), contradicting the outerplanarity of the part.

Lemma 7 formalizes the equivalence: G admits a proper k‑edge‑coloring iff G′ has outerthickness at most k. Consequently, deciding outerthickness for k ≥ 3 is at least as hard as the edge‑coloring problem, establishing NP‑hardness. The authors also verify that the problem lies in NP: a certificate is simply the edge partition, and each part can be tested for outerplanarity in linear time using Mitchell’s algorithm.

The paper then generalizes the construction to arbitrary classes F satisfying the three conditions. The same gadget H is chosen to be edge‑maximal with respect to F, and the labeling and attachment scheme are adapted so that the same forcing arguments apply, using a generalized version of Observation 6 that works for any F closed under topological minors and 1‑sums. This yields the full statement of Theorem 3.

Finally, the authors discuss the necessity of the three conditions. They provide counterexamples: if (a) is dropped, the class of Eulerian graphs yields a polynomial‑time solvable P_F; if (b) is dropped, pseudoforests give a polynomial algorithm; if (c) is dropped, forests are trivially solvable. Thus each condition is essential.

In summary, the paper delivers a concise, self‑contained proof that outerthickness is NP‑hard for any fixed k ≥ 3, extends this hardness to a broad family of graph covering problems, and clarifies the exact structural requirements on the target class F that make the problem computationally intractable. The reduction is elegant, relying on classical edge‑coloring hardness and a carefully designed outerplanar (or F‑maximal) gadget, and the work closes a notable gap in the literature on graph drawing and decomposition problems.