论文标题
部分可观测时空混沌系统的无模型预测
Further contributions on the outer multiset dimension of graphs
论文作者
论文摘要
图$ g $的外部多尺寸$ {\ rm dim} _ {\ rm ms}（g）$是最小的一组顶点的基数，它们通过使用该集合的多距离唯一地识别了此集合之外的所有顶点。证明$ {\ rm dim} _ {\ rm ms}（g）= n（g） - 1 $，仅当$ g $是直径的常规图，最多为$ 2 $。 $ {\ rm dim} _ {\ rm ms}（g）= 2 $的图形$ g $在多项式时间中描述并识别。当$ g $和$ h $的词典产物的下限是$ h $完整或无进取的，并且确定了极端图。证明$ {\ rm dim} _ {\ rm ms}（p_s \，\ square \，p_t）= 3 $ for $ s \ ge t \ ge ge 2 $。
The outer multiset dimension ${\rm dim}_{\rm ms}(G)$ of a graph $G$ is the cardinality of a smallest set of vertices that uniquely recognize all the vertices outside this set by using multisets of distances to the set. It is proved that ${\rm dim}_{\rm ms}(G) = n(G) - 1$ if and only if $G$ is a regular graph with diameter at most $2$. Graphs $G$ with ${\rm dim}_{\rm ms}(G)=2$ are described and recognized in polynomial time. A lower bound on the lexicographic product of $G$ and $H$ is proved when $H$ is complete or edgeless, and the extremal graphs are determined. It is proved that ${\rm dim}_{\rm ms}(P_s\,\square\, P_t) = 3$ for $s\ge t\ge 2$.
