关于互补棱镜的一般位置数
On the general position number of complementary prisms
摘要：
图$G$的一般位置数${\rm gp}(G)$是最大顶点集$S$的基数，使得$S$中没有任何元素位于$S$中其他两个元素之间的测地线上。$G$的互补棱柱$G\overline{G}$是由$G$和它的互补体$\overline{G}$的不交并通过添加它们之间完美匹配的边而形成的图。证明了当$G$连通时为${\rm gp}(G\overline{G})\le n(G) + 1$，当$G$不连通时为${\rm gp}(G\overline{G})\le n(G)$.图$G$的特征在于，如果$G$和$\overline{G}$都连通，则${\rm gp}(G\overline{G}) = n(G) + 1$对其成立。证明了${\rm gp}(G\overline{G})$的一个精确下界.若$G$是连通二部图或分裂图，则有${\rm gp}(G\overline{G})\in \{n(G), n(G)+1\}$。刻画了${\rm gp}(G\overline{G})=n(G)+1$成立的连通二部图和块图.本文构造了一类块图，其中它们的互补棱柱的${\rm gp}$ -数任意小于它们的阶数。
英文摘要：
The general position number ${\rm gp}(G)$ of a graph $G$ is the cardinality of a largest set of vertices $S$ such that no element of $S$ lies on a geodesic between two other elements of $S$. The complementary prism $G\overline{G}$ of $G$ is the graph formed from the disjoint union of $G$ and its complement $\overline{G}$ by adding the edges of a perfect matching between them. It is proved that ${\rm gp}(G\overline{G})\le n(G) + 1$ if $G$ is connected and ${\rm gp}(G\overline{G})\le n(G)$ if $G$ is disconnected. Graphs $G$ for which ${\rm gp}(G\overline{G}) = n(G) + 1$ holds, provided that both $G$ and $\overline{G}$ are connected, are characterized. A sharp lower bound on ${\rm gp}(G\overline{G})$ is proved. If $G$ is a connected bipartite graph or a split graph then ${\rm gp}(G\overline{G})\in \{n(G), n(G)+1\}$. Connected bipartite graphs and block graphs for which ${\rm gp}(G\overline{G})=n(G)+1$ holds are characterized. A family of block graphs is constructed in which the ${\rm gp}$-number of their complementary prisms is arbitrary smaller than their order.
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