Is the intersection of two intervals again an interval?
Solution 1:
No, it is not in general. Consider the bipartite graph $K_{2,4}$, with $\{a_1,b_1,a_2,b_2\}$ being the part of cardinal $4$ and $\{u,v\}$ the other part. Then $[a_1,b_1] \cap [a_2,b_2] = \{u,v\}$ is not an interval.