关于不含长导出路的图的一个注记(英文)

A Note on Graphs without Long Induced Paths

一个不含具有ｔ个顶点的导出路的图被称为是Ｐｔ－自由的．一个连通图Ｇ的ｉ－中心是由Ｖ（Ｇ）中所有距其它任何顶点的距离不大于ｉ的顶点组成的集合．对于Ｖ（Ｇ）的两个子集Ｓ和Ｔ，如果对任何ｘ∈Ｔ都有ｙ∈Ｓ，使得ｘ距ｙ的距离不大于ｄ，则称Ｓｄ－支配Ｔ．本文解决了由Ｏ．Ｆａｖａｒｏｎ和Ｊ．Ｌ．Ｆｏｕｑｕｅｔ提出的一个公开问题，即证明了如下结果：对任何Ｐｔ－自由图Ｇ，如果ｉ｜ｔ／２｜且ｐ１，则Ｃｉ（Ｇ）（ｐ＋１）－支配Ｃｉ＋ｐ．

A graph is said to be P t free whenever it dose not contain an induced path on t vertices.The i center of a connected graph G is the set of vertices whose distance from any other vertex is at most i. A subset S of V(G) d dominates a subset T of V(G) if for every vertex x∈T there is a vertex y∈S whose distance from x is at most d. In this note,it is proved that in any connected P t free graph G, C i+p (G) is (p+1) dominated by C i(G) for every i|t/2| and p1, which was proposed by O.Favaron and J.L.Fouquet as an open problem.