Excluded–Minor Characterizations of Antimatroids arisen from Posets and Graph Searches

An antimatroid is a family of sets such that it contains an empty set, and it is accessible and closed under union of sets. An antimatroid is a ’dual’ or ‘antipodal’ concept of matroid. ll show that an antimatroid is derived from shelling of a poset if and only if it does not contain a minor isomorphic to S7 where S7 is the smallest semimodular lattice that is not modular (See Fig. 1). It is also shown that an antimatroid is a node-search antimatroid of a digraph if and only if it does not contain a minor isomorphic to D5 where D5 is a lattice consisting of five elements Ø {x},{y}, {x, y} and {x, y, z}. Furthermore, an antimatroid is shown to be a node-search antimatroid of an undirected graph if and only if it does not contain D5 nor S10 as a minor: S10 is a locally free lattice consisting of ten elements shown in Fig. 2.