论文标题
一般$ D $ - 位置集
General $d$-position sets
论文作者
论文摘要
图$ g $的一般$ d $ - 位数$ {\ rm gp} _d（g）$是最大的集合$ s $的基数，对于最多的$ d $，$ s $的三个不同的顶点都不是$ s $。该新图参数概括了所研究的一般位置编号。我们首先给出一些有关$ d $的合适值的$ {\ rm gp} _d（g）$的单调行为的结果。我们表明，有关查找$ {\ rm gp} _d（g）$的决策问题对于任何值的$ d $而言都是NP complete。 $ {\ rm gp} _d（g）$的值当$ g $是一条路径或计算周期时，显示了一般$ d $ - 位置集的结构表征。此外，我们与其他主题提出了一些关系，包括强大的解决图和解离集。我们通过证明$ {\ rm gp} _d（g）$是无限的，只要$ g $是无限的图表，而$ d $是有限的整数，我们就可以完成论述。
The general $d$-position number ${\rm gp}_d(G)$ of a graph $G$ is the cardinality of a largest set $S$ for which no three distinct vertices from $S$ lie on a common geodesic of length at most $d$. This new graph parameter generalizes the well studied general position number. We first give some results concerning the monotonic behavior of ${\rm gp}_d(G)$ with respect to the suitable values of $d$. We show that the decision problem concerning finding ${\rm gp}_d(G)$ is NP-complete for any value of $d$. The value of ${\rm gp}_d(G)$ when $G$ is a path or a cycle is computed and a structural characterization of general $d$-position sets is shown. Moreover, we present some relationships with other topics including strong resolving graphs and dissociation sets. We finish our exposition by proving that ${\rm gp}_d(G)$ is infinite whenever $G$ is an infinite graph and $d$ is a finite integer.
