Better Bridges
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With the listed set of rules, none of the four puzzles are solvable. You need to "upgrade" some of the rules in order to be able to solve them. 

red  Puzzles with red borders are allowed to have triple bridges. 
green  Puzzles with green borders are allowed to have 45degree diagonal bridges. 
blue  Puzzles with blue borders are allowed to have bridges that intersect with each other. 
extraction  Which bridges used the modified rules? Those bridges will give you the answer. 
Rules of Hashi:
Connect the circles (“islands”) with straight bridges such that all the circles form one connected component, and
Connect the circles (“islands”) with straight bridges such that all the circles form one connected component, and
 At most two bridges connect each pair of islands;
 All bridges are horizontal or vertical;
 No two bridges intersect each other.