Discrete Mathematics ( IF 0.7 ) Pub Date : 2021-02-17 , DOI: 10.1016/j.disc.2021.112326 Steve Seif
Let be a non-trivial simple graph with vertices and edges , and let . Computing the diameter of and the min–max center of () are both quadratic-time (). A problem is strongly subquadratic-time if it is for some . If either the diameter problem or the center problem is strongly subquadratic-time, then the Strong Exponential Time Hypothesis would be violated. The same is true even for chordal graphs (graphs having no induced -cycle for ). This paper presents an algorithm that is faster than existing algorithms for the diameter problem for all chordal graphs.
With the size of a largest independent vertex subset of , it is proven here that the diameter problem for chordal graphs is -time. The algorithm does not require knowledge of ; nevertheless, it relates the diameter problem to the structure of .
Large-radius chordal graphs are likely to have small , which suggests the existence of interesting classes of large-radius chordal graphs having strongly subquadratic-time diameter problems.
中文翻译:
与中心问题有关的弦图的更快直径问题算法
让 是具有顶点的非平凡简单图 和边缘 , 然后让 。计算直径 和的最小最大中心 ()都是二次时间()。问题是强烈的次二次-时间,如果它是 对于一些 。如果直径问题或中心问题是强次时间,那么将违反强指数时间假说。即使对于弦图也是如此(没有归纳的图-周期 )。针对所有弦图的直径问题,本文提出了一种比现有算法更快的算法。
和 最大独立顶点子集的大小 ,在这里证明弦图的直径问题是 -时间。该算法不需要以下知识; 然而,它将直径问题与。
大半径弦图 可能会很小 ,这表明存在有趣的大半径弦图,它们具有强烈的次二次时间直径问题。