#CJam, 90 bytes

This is based on a iterative flood fill explained here and can be golfed a lot!

q~:Q{:IQ3/S*Sca5*+:T;G,G*{:AT=1$={[WXZ5 4_~_)_)]Af+Tf=AT='#a+&,g{TA'#t:T;}*}*}%;aT\/,3<},p 

Requires the input in order of ABCDEFGH like:

[0 1 1 2 1 0 5 5 1] 

and output is the connected nodes:

[1 1 2 1 5 5 1] 

Brief Explanation

• First, I iterate over the input array for each label.
• In each iteration, I perform floodfill to figure out disconnected nodes.
• If the number of disconnected nodes is greater than 1, then that label is disconnected.
• 1 because a label can have just 1 occurrence in the input array too.
• Then I simply filter out disconnected labels and print the array.

Full explanation to follow

Try it online here

#CJam, 90 bytes

This is based on a iterative flood fill explained here and can be golfed a lot!

q~:Q{:IQ3/S*Sca5*+:T;G,G*{:AT=1$={[WXZ5 4_~_)_)]Af+Tf=AT='#a+&,g{TA'#t:T;}*}*}%;aT\/,3<},p 

Requires the input in order of ABCDEFGH like:

[0 1 1 2 1 0 5 5 1] 

and output is the connected nodes:

[1 1 2 1 5 5 1] 

Brief Explanation

• First, I iterate over the input array for each label.
• In each iteration, I perform floodfill to figure out disconnected nodes.
• If the number of disconnected nodes is greater than 1, then that label is disconnected.
• 1 because a label can have just 1 occurrence in the input array too.
• Then I simply filter out disconnected labels and print the array.

Full explanation to follow

Try it online here

#CJam, 92 bytes

This is based on a recursive flood fill explained here and can be golfed a lot!

q~:Q_&{:IQ3/S*Sca5*+:T;G,G*{:AT=1$={[WXZ5 4_~_)_)]Af+Tf=AT='#a+&,g{TA'#t:T;}*}*}%;aT\/,3<},p 

Requires the input in order of ABCDEFGH like:

[0 1 1 2 1 0 5 5 1] 

and output is the connected nodes:

[1 2 5] 

Full explanation to follow

Try it online here

#CJam, 90 bytes

This is based on a iterative flood fill explained here and can be golfed a lot!

q~:Q{:IQ3/S*Sca5*+:T;G,G*{:AT=1$={[WXZ5 4_~_)_)]Af+Tf=AT='#a+&,g{TA'#t:T;}*}*}%;aT\/,3<},p 

Requires the input in order of ABCDEFGH like:

[0 1 1 2 1 0 5 5 1] 

and output is the connected nodes:

[1 1 2 1 5 5 1] 

Brief Explanation

• First, I iterate over the input array for each label.
• In each iteration, I perform floodfill to figure out disconnected nodes.
• If the number of disconnected nodes is greater than 1, then that label is disconnected.
• 1 because a label can have just 1 occurrence in the input array too.
• Then I simply filter out disconnected labels and print the array.

Full explanation to follow

Try it online here