One outstanding path from A to B

In the Norwegian fairy tale Three Billy Goats Gruff, the eponymous billy goats face a serious dilemma: the field they call home is barren and they must cross a bridge to reach greener pastures. The problem is that a terrible troll lives under the bridge and threatens to swallow the goats if they get too close. The troll’s authority over the only viable path between the two fields gives it enormous power over who comes and goes. After all, the billy goats, who ultimately outsmart the troll, would surely have taken a different path if one had been available. Seeking to capture the importance of outstanding paths, a network scientist would say that the troll has high betweenness centrality.