The online graph exploration problem, which was proposed by Kalyanasundaram and Pruhs (Theor Comput Sci 130(1):125–138, 1994), is defined as follows: Given an edge-weighted undirected connected graph and a specified vertex (called the origin), the task of an algorithm is to compute a path from the origin to the origin which contains all the vertices of the given graph. The goal of the problem is to find such a path of minimum weight. At each time, an online algorithm knows only the weights of edges each of which consists of visited vertices or vertices adjacent to visited vertices. Fritsch (Inform Process Lett 168:1006096, 2021) showed that the competitive ratio of an online algorithm is at most three for any unicyclic graph. On the other hand, Brandt et al. (Theor Comput Sci 839:176–185, 2020) showed a lower bound of two on the competitive ratio for any unicyclic graph. In this paper, we showed the competitive ratio of an online algorithm is at most 5/2 for any unicyclic graph.

在线图探索问题由 Kalyanasundaram 和 Pruhs 提出（Theor Comput Sci 130(1):125–138, 1994），定义如下：给定一个边加权无向连通图和一个指定的顶点（称为原点） ），算法的任务是计算从原点到包含给定图的所有顶点的原点的路径。问题的目标是找到这样一条权重最小的路径。每次，在线算法只知道边的权重，每条边由访问过的顶点或与访问过的顶点相邻的顶点组成。 Fritsch (Inform Process Lett 168:1006096, 2021) 表明，对于任何单环图，在线算法的竞争比最多为 3。另一方面，布兰特等人。 （Theor Comput Sci 839:176–185, 2020）显示任何单环图的竞争比的下限为 2。在本文中，我们证明了对于任何单环图，在线算法的竞争比最多为 5/2。