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：我们开发了一种线性时间算法，用于寻找无小行星三元组（AT-free）图的直径。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，在一般图中计算单纯顶点的问题可以通过现有规约转化为在AT-free图中寻找直径的问题，并且可以在O(n²)内完成。我们改进了几个图论问题的最佳已知时间复杂度。