Given a set of paths $P$ we define the \emph{Path Covering with Forest Number} of $P$} (PCFN($P$)) as the minimum size of a set $F$ of forests satisfying that every path in $P$ is contained in at least one forest in $F$. We show that PCFN($P$) is treatable when $P$ is a set of non-crossing shortest paths in a plane graph or subclasses. We prove that if $P$ is a set of non-crossing shortest paths of a planar graph $G$ whose extremal vertices lie on the same face of $G$, then PCFN($P$)\leq 4$, and this bound is tight.
翻译:鉴于一套路径,我们将每条路径(P$)确定为每条路径(PP$)(PCNFN(PP$))的最低大小,以满足每条路径(P$)都包含在至少一个森林(F$)中。我们证明,当每条路径(P$)是平面图或小类中一套非跨越最短路径时,每条路径(P$)是可以处理的。我们证明,如果每条路径(P$)是一套非跨越最短路径的平面图($G$),其底部脊椎位于同一面(G$)上,然后是每条底部脊椎(PP$)\leq 4美元,而这一界限是紧紧的。