You can delete the search shared with you by other users.
Procedure
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From the My Applications page in the Web Console, go to Monitoring > Log Monitoring.
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At the top of the page, click Search.
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Below the search bar, click Show History to show the Shared Search table.
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On the Shared Search page, click Shared To Me to list the shared searches.
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Click the appropriate shared search from the displayed list.
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In the upper-right of the Search page, click Actions > Unshare.
Results
The shared search is deleted from the users Shared To Me table.