# Online Graph Exploration on a Restricted Graph Class: Optimal Solutions for Tadpole Graphs

## TL;DR

This paper proves that a greedy algorithm achieves an optimal competitive ratio of 2 for online exploration of tadpole graphs with arbitrary edge weights, improving understanding of exploration efficiency on this graph class.

## Contribution

It demonstrates that a simple greedy approach is optimally competitive for exploring tadpole graphs with non-negative edge weights.

## Key findings

- Greedy algorithm achieves a competitive ratio of 2.
- Optimal exploration strategy for tadpole graphs established.
- Results extend previous bounds to non-unit edge weights.

## Abstract

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 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. Miyazaki et al. (The online graph exploration problem on restricted graphs, IEICE Transactions 92-D (9), 2009) showed that every online algorithm on these graphs must have a competitive ratio of 2-epsilon, but did not provide upper bounds for 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.