The longest and Hamiltonian path problems are well-known NP-hard problems in graph theory. Despite many applications of these problems, they are still open for many classes of graphs, including solid grid graphs and grid graphs with some holes. We consider the longest and Hamiltonian $(s,t)$-path problems in $C$-shaped grid graphs. A $(s,t)$-path is a path between two given vertices $s$ and $t$ of the graph. A $C$-shaped grid graph is a rectangular grid graph such that a rectangular grid subgraph is removed from it to make a $C$-liked shape. In this paper, we first give the necessary conditions for the existence of Hamiltonian cycles and Hamiltonian $(s,t)$-paths in such graphs. Then by given a linear-time algorithm for finding Hamiltonian cycles and Hamiltonian $(s,t)$-paths, we show that these necessary conditions are also sufficient. Finally, we give a linear-time algorithm for finding the longest $(s,t)$-path in these graphs.

最长路径和哈密顿路径问题是图论中众所周知的NP难题。尽管这些问题有许多应用，但它们仍可用于许多类的图形，包括实体网格图和带孔的网格图。我们考虑最长的哈密顿量$（s，Ť）$路径问题 $C$形的网格图。一种$（s，Ť）$-path是两个给定顶点之间的路径 $s$ 和 $Ť$图的 一种$C$形网格图是一个矩形网格图，以便从其中删除一个矩形网格子图以形成一个 $C$喜欢的形状。在本文中，我们首先给出存在哈密顿循环和哈密顿量的必要条件$（s，Ť）$-此类图中的路径。然后通过给出线性时间算法来查找哈密顿循环和哈密顿$（s，Ť）$-paths，我们证明这些必要条件也是足够的。最后，我们给出一个线性时间算法来寻找最长的$（s，Ť）$这些图中的-path。