Determining the Outerthickness of Graphs Is NP-Hard

TL;DR

The paper resolves the open problem of outerthickness for general graphs by proving $OuterThickness(G,k)$ is NP-complete for every fixed $k≥3$. The proof builds a gadget-based reduction from Edge-Coloring on k-regular graphs, using a labeling mechanism and an auxiliary graph to enforce color constraints so that a valid k-edge-coloring corresponds to a low-thickness partition. The authors generalize the reduction to a covering problem $P_F$ for any decidable proper class F closed under topological minors and 1-sums and containing a triangle, showing NP-hardness and NP membership when F membership is poly-time decidable. Consequences include NP-hardness results for outerplanar and planar thickness (complementing Mansfield's result for k=2) and a demonstration that the three conditions on F are necessary, providing a unified hardness framework for thickness-type problems. The work resolves a long-standing open question and offers a broadly applicable reduction technique for edge-coloring to thickness-type parameters.

Abstract

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.

Figures (2)

• Figure 1: An illustration of the construction of $G'$. For each edge $e\coloneqq\{u,v\}\in E(G)$, we add the edges $\{u,w_{\varphi(e)}\}$ and $\{v,w_{\varphi(e)}\}$ to $G'$. In this example, we have $\mathcal{L} = \{1, 2, \ldots, 6\}$, and we depict the added edges corresponding to $e$ with $\varphi(e) \in \{1,2\}$.
• Figure 2: An illustration of $S_{y,y'}$ with $Q = K_4 -\{e\}$ and $z =1$.

Theorems & Definitions (9)

• Theorem 2
• Theorem 3
• Corollary 4
• Theorem 5: Edge-Coloring Holyer81LevenG83
• Lemma 7
• Proposition 11
• Proposition 12
• Lemma 13
• Lemma 16