Edge-decomposition into Two Triangular Forests is NP-complete
2026-07-15 • Computational Complexity
Computational ComplexityDiscrete Mathematics
AI summaryⓘ
The authors study a problem where you want to split the edges of a graph into two parts, each part forming a special kind of graph called a triangular forest (made of triangles connected in a simple way). Previous work showed that splitting edges into three or more such parts is very hard to solve (NP-hard), but cases with forests (no triangles) are easier. The authors prove that even splitting into just two triangular forests is still hard (NP-complete). This helps understand which edge-splitting problems are computationally difficult.
graph classedge-decompositiontopological minors1-sumstriangular forestsNP-completeouterplanar graphs2-connected componentscomputational complexity
Authors
Beniamin Bibrowski, Tomáš Masařík
Abstract
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.