We study the problem of exploring all vertices of an undirected weighted graph that is initially unknown to the searcher. An edge of the graph is only revealed when the searcher visits one of its endpoints. Beginning at some start node, the searcher's goal is to visit every vertex of the graph before returning to the start node on a tour as short as possible.

We prove that the Nearest Neighbor algorithm's competitive ratio on trees with n vertices is $\mathrm{\Theta }(\log n)$, i.e. no better than on general graphs. Furthermore, we examine the algorithm Blocking for a range of parameters not considered previously and prove it is 3-competitive on unicyclic graphs as well as $5/2+\sqrt{2}\approx 3.91$-competitive on cactus graphs. The best known lower bound for these two graph classes is 2.

我们研究了探索搜索者最初不知道的无向加权图的所有顶点的问题。仅当搜索者访问其端点之一时，才显示图形的边缘。从某个起始节点开始，搜索者的目标是在尽可能短的时间内返回到起始节点之前，先访问图的每个顶点。

我们证明了最近邻算法在具有n个顶点的树上的竞争比为$\mathrm{\Theta }（\mathrm{日志}ñ）$，即不比一般图形更好。此外，我们针对先前未考虑的参数范围检查了算法Blocking，并证明它在单环图以及$5/2+\sqrt{2}\approx 3.91$-在仙人掌图上具有竞争力。这两个图类的最著名下限是2。