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The reasons the detective suspect Mr. Barker and Mrs. Douglas because:
1) In Act 2 scene 3 Watson sees that Mr. Barker and Mrs. Douglas are sitting on a bench, enjoying a lively conversation. Mrs. Douglas was laughing and was happy. This was not correct, if someone's husband dies the wife should cry and should be sad but Mrs. Douglas was happy so by looking at this it looks like Mrs. Douglas has killed her husband.
2) Mr. Douglas did not like the relationship between Mr. Barker and Mrs. Douglas. So Mr. Barker and Mrs. Douglas could have killed Mr. Douglas so that no one has a problem with their relationship.