Alternative to relative clauses

The rule is that if the relative pronoun refers back to the subject of the sentence, it may not be omitted, but if it refers back to the object, it may.

There is one exception to the above rule: if placing the relative clause right after "its" noun in this way would leave the verb in the main clause dangling at the end of the sentence by itself, the resulting sentence would be awkward to comprehend. In such cases, the verb is usually moved in front of the relative clause.

The man who is wearing the glasses looked at me.

 The man wearing the glasses looked at me

The house that is on the corner is mine

The house on the corner is mine.

A relative pronoun that, who, which + be  (is, are, was, were...) can be deleted from a modifiying clause to shorten it to a participial phrase. 

• active verbs – change to a modifier ending in -ING   (present participle form)
• passive verbs – change to a modifier ending in -ED (past participle form)

There is an exception

The tense of be is needed to make sense.   

 The London Bridge [that was] located in England is now in Arizona.

The London Bridge  located in England is now in Arizona.   (It is not clear that the bridge was relocated or rebuilt in a new place.)

The phrase still makes sense without the be verb.   

The London Bridge  [that is] located in Arizona used to be in England