边分解为两个三角森林是NP完全问题
Edge-decomposition into Two Triangular Forests is NP-complete
AI总结：
研究将图边分解为两个三角森林问题的复杂度，通过考虑满足特定条件的最简单图类三角森林，证明该边分解问题是NP完全的。
AI中文摘要：
设$\mathcal F$是一个在拓扑子图和1和运算下封闭、成员资格可判定、包含三角形且不是所有图的类的图类。Lee等人表明将图边分解为$\mathcal F$的$k \geq 3$个元素的问题是NP难的。本文研究$k = 2$的情况，考虑满足上述标准的最简单图类$\mathcal F$——三角森林，即每个2连通分量都是三角形的图。证明判定一个图是否能边分解为两个三角森林是NP完全的。
英文摘要：
Let $\mathcal F$ be a graph class that is closed under topological minors and 1-sums, has decidable membership, contains a triangle, and is not the class of all graphs. Recently, Lee, Liu, and Tsai [ICALP 2026] showed that the edge-decomposition problem into $k \geq 3$ elements of $\mathcal F$ is NP-hard. In particular, their general hardness reduction covers a long-standing problem on outerthickness (when $\mathcal F$ is the class of outerplanar graphs). On the other hand, it is well known that decomposing a graph into forests is polynomial-time solvable, as implied by work of Edmonds [J. Res. Natl. Bur. Stand. B. 1965]. In this paper, we take a first step toward determining the complexity of edge-decomposition problems into just two graphs (the case $k=2$). We consider the simplest possible graph class $\mathcal F$ satisfying the criteria above: the triangular forests, that is, graphs in which every 2-connected component is a triangle. We prove that determining whether a graph can be edge-decomposed into two triangular forests is NP-complete.
