We develop a linear time algorithm for finding the diameter of an asteroidal triple-free (AT-free) graph. Furthermore, we update the definition of polar pairs and develop new properties of polar pairs for (weak) dominating pair graphs. We prove that the problem of computing a simplicial vertex in a general graph can be accomplished in O(n^2) based on an existing reduction to the problem of finding diameter in an AT-free graph. We improve the best-known run-time complexities of several graph theoretical problems.

翻译：我们开发了一种用于寻找无星形三元组（AT-free）图直径的线性时间算法。此外，我们更新了极对的定义，并为（弱）支配对图建立了极对的新性质。我们证明，基于现有将一般图中计算单纯顶点问题归约到AT-free图直径问题的技术，该问题可在O(n²)时间内完成。我们改进了若干图论问题的最优已知运行时间复杂度。