We introduce the following notion of compressing an undirected graph G with (nonnegative) edge-lengths and terminal vertices R???V(G). A distance-preserving minor is a minor G' (of G) with possibly different edge-lengths, such that R???V(G') and the shortest-path distance between every pair of terminals is exactly the same in G and in G'. We ask: what is the smallest f ~*(k) such that every graph G with k?=?|R| terminals admits a distance-preserving minor G' with at most f ~*(k) vertices? Simple analysis shows that f ~*(k)?≤?O(k ~4). Our main result proves that f ~*(k)?≥?Ω(k ~2), significantly improving over the trivial f ~*(k)?≥?k. Our lower bound holds even for planar graphs G, in contrast to graphs G of constant treewidth, for which we prove that O(k) vertices suffice.
展开▼
机译：我们介绍了用（非负）边长度和终端顶点r ??? V（g）压缩无向图g的以下概念。距离保留的次要是一个次要的g'（g），具有可能不同的边缘长度，使得每对终端之间的r ??? V（g'）和g之间的最短路径距离在g和在g'中。我们问：什么是最小的f〜*（k），使得每个图表g与k？=？| r |终端承认最多F〜*（k）顶点的距离保留的小g'？简单的分析表明，F〜*（k）？≤αo（k〜4）。我们的主要结果证明，F〜*（k）？≥≤Ω（k〜2），显着改善了微小的f〜*（k）？≥？k。我们的下限甚至保持平面图G，与恒定树宽的图表G相反，我们证明O（k）顶点足够了。
展开▼
