Also appears in collection : ALEA Days 2018 / Journées ALEA 2018

Graph searching, a mechanism to traverse a graph visiting one vertex at a time in a specific manner, is a powerful tool used to extract structure from various families of graphs. In this talk, we focus on two graph searches: Lexicographic Breadth First Search (LBFS), and Lexicographic Depth First Search (LDFS). Many classes of graphs have a vertex ordering characterisation, and we review how graph searching is used to produce efficiently such vertex orderings. These orderings expose structure that we exploit to develop efficient linear and near-linear time algorithms for some NP-hard problems (independent set, colouring, Hamiltonicity for instance) on some special classes of graphs such as cocomparability graphs. In particular, we will prove fixed point type theorems for LexBFS, and then focus on a LexDFS-based framework to lift algorithms from interval graphs to cocomparability graphs. Then I will present the relationships between graph searches, graph geometric convexities and antimatroids. These relationships are for to be completely understood and I will pose some hard conjectures and some interesting problems to consider. To finish I will present some recent results about Robinsonian matrices by M. Laurent and M. Seminaroti and their relationships with graph searches. This yields a new area of research to investigate.