We study the problem of online graph exploration on undirected graphs, where a searcher has to visit every vertex and return to the origin. Once a new vertex is visited, the searcher learns of all neighboring vertices and the connecting edge weights. The goal of such an exploration is to minimize its total cost, where each edge traversal incurs a cost of the corresponding edge weight.

We investigate the problem on tadpole graphs (also known as dragons, kites), which consist of a cycle with an attached path. The construction by Miyazaki et al. (The online graph exploration problem on restricted graphs, IEICE Transactions 92-D (9), 2009) can be extended to show that every online algorithm on these graphs must have a competitive ratio of $2-\epsilon$, but the authors did not investigate non-unit edge weights. We show via amortized analysis that a greedy approach yields a matching competitive ratio of 2 on tadpole graphs, for arbitrary non-negative edge weights. Moreover, we also briefly discuss the topic of advice complexity on cycle and tadpole graphs.

我们研究了在无向图上进行在线图探索的问题，其中搜索者必须访问每个顶点并返回原点。一旦访问了新的顶点，搜索器就会了解所有相邻的顶点和连接边权重。这种探索的目的是最小化其总成本，其中每次边缘遍历都会产生相应边缘权重的成本。

我们在t图（也称为龙，风筝）上研究该问题，t图包括一个带有附加路径的循环。宫崎骏等人的建筑。（限制图上的在线图探索问题，IEICE Transactions 92-D（9），2009）可以扩展为显示这些图上的每个在线算法必须具有的竞争比为$2-\epsilon$，但作者没有研究非单位边缘权重。我们通过摊销分析表明，对于任意非负边缘权重，贪婪方法在t图上产生匹配的竞争比2。此外，我们还简要讨论了关于循环图和t图的建议复杂性主题。