I have 18 students in my graph theory course this semester: Anne, Bernard, Clare, David,..., and Rachel. At the start of the course I asked them to draw the graph below, in which each of them is represented by a vertex, two of which are joined by a line if, and only if, they represent two students who are friends.
Fewer than half my students turned up for class last Friday. However, because each of the absent students happened to be friends with at least one of those who did attend, I was able to return everyone's assignments either personally or through a friend. In fact, had a smaller group of students shown up for my class on Friday, I would have been unable to do this.