Edge-decomposition into Two Triangular Forests is NP-complete

2026-07-15

AI Summary

This study addresses the computational complexity of determining whether the edge set of a graph can be partitioned into two triangle forests—graphs in which every 2-connected component is a triangle. By constructing a polynomial-time reduction from a known NP-complete problem and leveraging structural properties of graphs, the authors establish for the first time that this edge decomposition problem is NP-complete when \( k = 2 \). This result resolves a critical gap in graph decomposition theory, bridging the boundary between polynomial-time solvability and NP-hardness. It thereby delineates the precise computational threshold for this class of decomposition problems and provides a foundational reference for the complexity classification of related graph decomposition questions.

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.

Problem

• edge-decomposition
• triangular forests
• NP-completeness
• graph decomposition
• computational complexity

Innovation

• edge-decomposition
• triangular forests
• NP-completeness
• graph classes
• computational complexity