In this paper, we show that determining whether a given graph admits a min-1-planar drawing is NP-hard. A drawing of a graph is called min-$k$-planar if, for every crossing in the drawing, at least one of the crossing edges involves at most $k$ crossings. This notion of min-$k$-planarity was introduced by Binucci et al. in 2023 as a generalization of $k$-planarity.
The proof of this result utilizes complex combinatorial methods, demonstrating the computational difficulty of the min-1-planarity problem. By deeply analyzing the structural features of graphs, the researchers reveal its NP-hardness and provide new directions for future research.
// Example code to check the min-1-planarity of a graph
bool isMin1Planar(Graph g) {
// Implement the specific checking logic
}Blogger's Review: This finding is significant for the field of graph theory, revealing the complexity of min-1-planarity, which may drive further research and development of related algorithms. Understanding the difficulty of such problems is crucial for optimizing graph drawing algorithms.