Consider a connected graph $G$ and let $T$ be a spanning tree of $G$. Every edge $e \in G-T$ induces a cycle in $T \cup \{e\}$. The intersection of two distinct such cycles is the set of edges of $T$ that belong to both cycles. The MSTCI problem consists in finding a spanning tree that has the least number of such non-empty intersections and the instersection number is the number of non-empty intersections of a solution. In this article we consider three aspects of the problem in a general context (i.e. for arbitrary connected graphs). The first presents two lower bounds of the intersection number. The second compares the intersection number of graphs that differ in one edge. The last is an attempt to generalize a recent result for graphs with a universal vertex.

翻译：考虑连通图$G$，令$T$为$G$的一棵生成树。$G-T$中的每条边$e$都会在$T \cup \{e\}$中诱导一个环。两个不同环的交集是同时属于这两个环的$T$中边的集合。MSTCI问题旨在找到一棵使此类非空交集数量最少的生成树，且交集数即为此解中非空交集的数量。本文在一般情境下（即针对任意连通图）探讨该问题的三个层面：第一层面给出交集数的两个下界；第二层面比较仅相差一条边的图的交集数；第三层面尝试推广近期关于存在泛化顶点的图的相关结论。