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{\bf Steven J. Tedford}
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{\bf Connectivity of the Lifts of a Greedoid}
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Recently, attempts were made to generalize the undirected branching
greedoid to a greedoid whose feasible sets consist of sets of edges
containing the root satisfying additional size restrictions. Although
this definition does not always result in a greedoid, the lift of the
undirected branching greedoid has the properties desired by the
authors.
The $k$-th lift of a greedoid has sets whose nullity is at most $k$ in
the original greedoid. We prove that if the greedoid is
$n$-connected, then its lift is also $n$-connected. Additionally, for
any cut-vertex $v$ and cut-edge $e$ of a graph $\Gamma$, let $C(v)$ be
the component of $\Gamma\setminus v$ containing the root and $C(e)$ be the
component of $\Gamma\setminus e$ containing the root. We prove that if the
$k$-th lift of the undirected branching greedoid is 2-connected, then
$$\eqalign{
|{E(C(v))}|&<|{V(C(v))}|+k-1\hbox{ and }\cr
|{E(C(e))}|&>|{E(\Gamma)}|-{k}-2.\cr
}$$
We also give examples indicating that no sufficient conditions for the
$k$th lift to be 2-connected exists similar to these necessary
conditions.
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