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Characterising AT-free Graphs with BFS
An asteroidal triple free graph is a graph such that for every independent
triple of vertices no path between any two avoids the third. In a recent result
from Corneil and Stacho, these graphs were characterised through a linear
vertex ordering called an AT-free order. Here, we use techniques from abstract
convex geometry to improve on this result by giving a vertex order
characterisation with stronger structural properties and thus resolve an open
question by Corneil and Stacho. These orderings are generated by a modification
of BFS which runs in polynomial time. Furthermore, we give a linear time
algorithm which employs multiple applications of (L)BFS to compute AT-free
orders in claw-free AT-free graphs and a generalisation of these