# Negative edge weight in Dijkstra

Suppose given an un-directed graph $$G$$, such that bridge edge of $$G$$ has negative weight.

From Wikipedia:

In graph theory, a bridge, isthmus, cut-edge, or cut arc is an edge of a graph whose deletion increases the graph's number of connected components.

Now there is a claim:

The Dijkstra find correct shortest simple-path if the bridge edge has negative weight.

I think this claim is correct, but i can't show it. Already i know that, if edges of source $$s$$ have negative weight, then Dijkstra can find correct shortest path.

• However you can walk from "side-to-side" on the edge, decreasing the total cost with each time. If the edge was directed, then its a different problem. Aug 1, 2021 at 12:36
• @Rostami.M The claim is false, as nir shahar pointed out. Also, it's a bit pedantic but notice that, in the claim, we are not promised anything about the weights of the other (non-bridge) edges so, for example, it is conceivable that all edge weights of $G$ are negative. Aug 1, 2021 at 12:39