1.2 Definitions: The authors define key concepts such as ban-the-bombs, long bombs, and three-in-a-tree problems. They also explain the notation used throughout the paper.
1.3 Overview: This section provides a brief summary of the main results and techniques used in the paper.
1.4 Related Work: The authors discuss previous work on detecting holes in graphs, including structural results and algorithmic approaches.
2 Detecting a Long Even Hole: This section describes the algorithm for detecting long even holes using the three-in-a-tree problem. They explain how to reduce the problem to finding a path of length 3 in a tree and provide a detailed description of the approach.
2.1 Structural Results: The authors present structural results related to the three-in-a-tree problem, which are used to guide the algorithm.
2.2 Detecting (odd, even) Holes: They describe how to detect holes that are not long even holes using a separate algorithm based on the three-in-a-tree problem.
In summary, this paper presents a new algorithm for detecting long even holes in graphs based on the three-in-a-tree problem. The authors provide a detailed description of their approach and demonstrate its effectiveness through theoretical results and experimental evaluations. They also acknowledge the support of their colleagues and institutions, as well as the helpful discussions with other researchers in the field.
