Before revision:  It also provides constant-time support for all the operations provided by different approaches in previous work, as well as new operations that allow us to retrieve the last internal node before or the firstinternal node after a given node in a level-order traversal . This new representation gives us the functionality needed to design the first succinct distance oracles for interval graphs, proper interval graphs and k-proper/k-improper interval graphs .
After revision:  It also provides constant-time support for all the operations provided by different approaches in previous work, as well as new operations that allow us to retrieve the last internal node before or the firstinternal node after a given node in a level-order traversal , all in constant time .