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If a lightning network transaction between 2 participants, where the payment is routed by multiple channels, does it means that all full LN nodes that routed the payment should stay online to avoid Fraudulent Channel Close ?

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No, a channel’s closing affects that channel only.

When routing payments, a channel’s state will be updated to account for that payment. When closing the channel, both participants will make sure that the most recent state is published.

This process does not involve any external channels/participants. Closing a channel will not affect the balance of any other channel, as each individual channel’s participants will have their own account of what that channel’s most recent state is. The closure of another channel does not affect this state, even if it is closed fraudulently.

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