  Let $\F$ denote a field, and let $V$ denote a vector space over $\F$ with
finite positive dimension. A Leonard pair on $V$ is an ordered pair of
diagonalizable $\F$-linear maps $A: V \to V$ and $A^* : V \to V$ that each act
on an eigenbasis for the other in an irreducible tridiagonal fashion. Let
$A,A^*$ denote a Leonard pair on $V$. Let $\{v_i\}_{i=0}^d$ denote an
eigenbasis for $A^*$ on which $A$ acts in an irreducible tridiagonal fashion.
For $0 \leq i \leq d$ define an $\F$-linear map $E^*_i : V \to V$ such that
$E^*_i v_i = v_i$ and $E^*_i v_j = 0$ if $j \neq i$ $(0 \leq j \leq d)$. The
map $F = \sum_{i=0}^d E^*_i A E^*_i$ is called the flat part of $A$. The
Leonard pair $A,A^*$ is bipartite whenever $F=0$. The Leonard pair $A,A^*$ is
said to be near-bipartite whenever the pair $A-F, A^*$ is a Leonard pair on
$V$. In this case, the Leonard pair $A-F, A^*$ is bipartite, and called the
bipartite contraction of $A,A^*$. Let $B,B^*$ denote a bipartite Leonard pair
on $V$. By a near-bipartite expansion of $B,B^*$ we mean a near-bipartite
Leonard pair on $V$ with bipartite contraction $B,B^*$. In the present paper we
have three goals. Assuming $\F$ is algebraically closed, (i) we classify up to
isomorphism the near-bipartite Leonard pairs over $\F$; (ii) for each
near-bipartite Leonard pair over $\F$ we describe its bipartite contraction;
(iii) for each bipartite Leonard pair over $\F$ we describe its near-bipartite
expansions.
