By making use of lexicographic breadth first search, Lex-BFS and partition refinement with pivots, we obtain very simple algorithms for some well-known problems in graph theory. We give an O(n + m log n) algorithm for transitive orientation of a comparability graph, and simple linear algorithms to recognize interval graphs, convex graphs, Y-semichordal graphs and matrices that have the consecutive-ones property. Previous approaches to these problems used difficult preprocessing steps such as computing PQ trees or modular decomposition. The algorithms we give are easy to understand and straightforward to prove. They do not make use of sophisticated data structures and the complexity analysis is straightforward.