path. We associate a *cross* $c:=(\ell_H, \ell_V)$ to $P$ in which $\ell_H$ and $\ell_V$ denote its horizontal and vertical segments, respectively; we call $\ell_H$ and $\ell_V$ the *supporting segments* of $c$. The left endpoint of $\ell_H$ is $x:=(x({\texttt{corner}}(P))-\delta, y({\texttt{corner}}(P)))$ and its right endpoint is $x':=((x({\texttt{hTip}}(P))-3\delta, y({\texttt{corner}}(P)))$ in which $\delta:=s/4$, where $s$ is the length of a grid edge. See the figure on right for an illustration. Similarly, the bottom endpoint of $\ell_V$ is $y:=(x({\texttt{corner}}(P), y({\texttt{corner}}(P))-\delta)$ and its top endpoint is $y':=((x({\texttt{corner}}(P)), y({\texttt{vTip}}(P))-3\delta)$. We denote the cross of $P$ by ${\texttt{cross}}(P)$. The following observation is immediate by the construction of a cross.

\[obs:crossAndPath\] Let $P_u, P_v$ be two paths in [$\mathcal{P}_{{\texttt{vtx}}}$ ]{}. Then, $P_u$ and $P_v$ intersect each other at a grid node if and only if ${\texttt{cross}}(P_u)$ and ${\texttt{cross}}(P_v)$ intersect each other.

To see the hitting elements of the set system, for each path $P\in$[$\mathcal{P}_{{\texttt{vtx}}}$ ]{}, we add into $\mathcal{U}$ both of the supporting segments of ${\texttt{cross}}(P)$. We say that a hitting element $e\in \mathcal{U}$ *hits* a cross $c$ if $e$ intersects one of the supporting segments of $c$; we assume that the hitting elements corresponding to supporting segments of $c$ also hit $c$. To compute $\mathcal{S}$, consider the set 