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

7 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 an ‘antipodal’ concept of matroid. 9 We shall 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 11 modular. It is also shown that an antimatroid is a node-search antimatroid of a rooted digraph if and only if it does not contains a minor isomorphic to D5 where D5 is a lattice consisting of 13 1ve elements ∅; {x}; {y}; {x; y} and {x; y; z}. Furthermore, it is shown that an antimatroid is a node-search antimatroid of a rooted undirected graph if and only if it does not contain D5 nor 15 S10 as a minor: S10 is a locally free lattice consisting of 10 elements. ? 2002 Published by Elsevier Science B.V. 17