  A vertex in a graph is simplicial if its neighborhood forms a clique. We
consider three generalizations of the concept of simplicial vertices: avoidable
vertices (also known as \textit{OCF}-vertices), simplicial paths, and their
common generalization avoidable paths, introduced here. We present a general
conjecture on the existence of avoidable paths. If true, the conjecture would
imply a result due to Ohtsuki, Cheung, and Fujisawa from 1976 on the existence
of avoidable vertices, and a result due to Chv\'atal, Sritharan, and Rusu from
2002 the existence of simplicial paths. In turn, both of these results
generalize Dirac's classical result on the existence of simplicial vertices in
chordal graphs.
  We prove that every graph with an edge has an avoidable edge, which settles
the first open case of the conjecture. We point out a close relationship
between avoidable vertices in a graph and its minimal triangulations, and
identify new algorithmic uses of avoidable vertices, leading to new
polynomially solvable cases of the maximum weight clique problem in classes of
graphs simultaneously generalizing chordal graphs and circular-arc graphs.
Finally, we observe that the proved cases of the conjecture have interesting
consequences for highly symmetric graphs: in a vertex-transitive graph every
induced two-edge path closes to an induced cycle, while in an edge-transitive
graph every three-edge path closes to a cycle and every induced three-edge path
closes to an induced cycle.
