Given a graph $G$, a geodesic packing in $G$ is a set of vertex-disjoint maximal geodesics, and the geodesic packing number of $G$, ${\gpack}(G)$, is the maximum cardinality of a geodesic packing in $G$. It is proved that the decision version of the geodesic packing number is NP-complete. We also consider the geodesic transversal number, ${\gt}(G)$, which is the minimum cardinality of a set of vertices that hit all maximal geodesics in $G$. While $\gt(G)\ge \gpack(G)$ in every graph $G$, the quotient ${\rm gt}(G)/{\rm gpack}(G)$ is investigated. By using the rook's graph, it is proved that there does not exist a constant $C < 3$ such that $\frac{{\rm gt}(G)}{{\rm gpack}(G)}\le C$ would hold for all graphs $G$. If $T$ is a tree, then it is proved that ${\rm gpack}(T) = {\rm gt}(T)$, and a linear algorithm for determining ${\rm gpack}(T)$ is derived. The geodesic packing number is also determined for the strong product of paths.
翻译:以 G$ 计, 以 G$ 计的大地学包装是一组顶端分解最大大地学的一组顶端, 而大地学包装编号为$G$, $\gpack}(G) 美元, 是以$G$计的大地学包装的最大基本内容。 事实证明, 大地学包装编号的决定版本是 NP 完成 。 我们还考虑到大地学横贯号, $\ gt} (G) 美元, 这是一组以$G$ 打击所有最大大地学的顶端的最基本内容。 $gt (G)\ gpack (G) $ ggpack (G) 美元, 而每张Ggpack (G) 美元 美元 的底值是 $Gm 。 美元 美元 的底值, 美元 美元