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Graphs are partitioned into six classes from the perspective of chirality, depending on whether they are topologically achiral, whether there is at least one topologically achiral embedding, whether there is at least one rigidly achiral embedding, and whether there is at least one rigidly achiral presentation. Three of these classes are well represented by a variety of chemical structures: topologically chiral molecular graphs with no topologically achiral embeddings, topologically chiral molecular graphs with at least one rigidly achiral embedding, and topologically achiral molecular graphs with at least one rigidly achiral presentation. Known representatives of these three classes are described. Various meanings associated with the concepts "molecular graph" and "intrinsic chirality" are critically discussed. Previous arrangements of molecular graphs and molecules in a hierarchical order, ranging from the most to the least chiral, are interpreted in terms of the graph's and molecule's "chiral persistence".

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