Characterization of removable elements with respect to having $k$ disjoint bases in a matroid

Abstract

The well-known spanning tree packing theorem of Nash-Williams and Tutte characterizes graphs with $k$ edge-disjoint spanning trees. Edmonds generalizes this theorem to matroids with $k$ disjoint bases. For any graph $G$ that may not have $k$-edge-disjoint spanning trees, the problem of determining what edges should be added to $G$ so that the resulting graph has $k$ edge-disjoint spanning trees has been studied by Haas (2002) [11] and Liu et al. (2009) [17], among others. This paper aims to determine, for a matroid $M$ that has $k$ disjoint bases, the set $E_k\left(M\right)$ of elements in $M$ such that for any $e\in E_k\left(M\right)$, $M-e$ also has $k$ disjoint bases. Using the matroid strength defined by Catlin et al. (1992) [4], we present a characterization of $E_k\left(M\right)$ in terms of the strength of $M$. Consequently, this yields a characterization of edge sets $E_k\left(G\right)$ in a graph $G$ with at least $k$ edge-disjoint spanning trees such that $\forall e\in E_k\left(G\right)$, $G-e$ also has $k$ edge-disjoint spanning trees. Polynomial algorithms are also discussed for identifying the set $E_k\left(M\right)$ in a matroid $M$, or the edge subset $E_k\left(G\right)$ for a connected graph $G$.