关于图的距离控制数的上界

对于任意的正整数l,连通图G的顶点子集D被称为距离l 控制集 ,是指对于任意顶点v D ,D中至少含有一个顶点u ,使得距离dG(u ,v) ≤l.图G距离l 控制数γl(G)是指G中所有距离l 控制集的基数的最小者 .确定图G的距离l 控制数γl(G)是NP 问题 .给出了当G是阶数为p (p ≥l 1 )的连通图时 ,对于任意的正整数l,都有最优上界γl(G)≤ p-Δ l - 1 l .而且针对某些Δ和l,是对Meir和Moon的结果的一种改进

Bounds for Distance Domination Numbers of Graphs

For any positive integer l and any graph G=(V,E), a set D of vertices of G is said to be an l-dominating set if every vertex in V(G)-D is at distance at most l from some vertex in D. The l-domination number, γl(G), is the minimum cardinality of an l-dominating set of G. For a given graph G and an integer l, to determine γl(G) is an NP-hard problem. It is proved in this paper that if G is a connected graph of order p with p≥ l+1, then γl(G)≤(p-Δ+l-1)/(l) for any positive integer l. This bound is the best possible for some Δ and l and is an improvement on some known results.