Mader conjectured in 2010 that for any tree T of order m, every k-connected graph G with minimum degree at least \(\lfloor \frac{3k}{2} \rfloor +m-1\) contains a subtree \(T' \cong T\) such that \(G-V(T')\) is k-connected. This conjecture has been proved for \(k = 1\); however, it remains open for general \(k \ge 2\); for \(k = 2\), partially affirmative answers have been shown, all of which restrict the class of trees to special subclasses such as trees with at most 5 internal vertices, trees of order at most 8, trees with diameter at most 4, caterpillars, and spiders. We first extend the previously known subclass of trees for which Mader’s conjecture for \(k = 2\) holds; namely, we show that Mader’s conjecture for \(k = 2\) is true for the class of bifurcate quasi-unimodal caterpillars which includes every caterpillar and every tree of order m with diameter at least \(m-4\). Instead of restricting the class of trees, we next consider 2-connected graphs with girth conditions. We then show that Mader’s conjecture is true for every 2-connected graph G with \(g(G) \ge \delta (G)-8\), where g(G) and \(\delta (G)\) denote the girth of G and the minimum degree of a vertex in G, respectively. Besides, we show that for every 2-connected graph G with \(g(G) \ge \delta (G)-7\), the lower bound of \(m+2\) on \(\delta (G)\) in Mader’s conjecture can be improved to \(m+1\) if \(m \ge 10\). Moreover, the lower bound of \(\delta (G)-8\) (respectively, \(\delta (G)-7\)) on g(G) in these results can be improved to \(\delta (G)-9\) (respectively, \(\delta (G) -8\) with \(m \ge 11\)) if no six (respectively, four) cycles of length g(G) have a common path of length \(\left\lceil \frac{g(G)}{2} \right\rceil -1\) in G. We also show that Mader’s conjecture holds for every 2-connected graph G with \(g^\circ (G) \ge \delta (G)-8\), where \(g^\circ (G)\) is the overlapping girth of G. Mader’s conjecture is interesting not only from a theoretical point of view but also from a practical point of view, since it may be applied to fault-tolerant problems in communication networks. Our proofs lead to \(O(|V(G)|^4)\) time algorithms for finding a desired subtree in a given 2-connected graph G satisfying the assumptions.

Mader在2010年推测，对于m阶的任何树T，每个最小度至少为\（\ lfloor \ frac {3k} {2} \ rfloor + m-1 \）的k个连通图G都包含一个子树\（T '\ cong T \）使得\（GV（T'）\）被k连接。这个猜想已经被证明为\（k = 1 \） ; 但是，它仍然对通用\（k \ ge 2 \）开放；对于\（k = 2 \）给出了部分肯定的答案，所有这些都将树的类别限制在特殊的子类中，例如具有最多5个内部顶点的树，有序的树最多8个，直径最大为4个的树，毛毛虫和蜘蛛。我们首先扩展先前已知的树的子类，其中对（\ k = 2 \）的Mader猜想成立。即，我们表明，Mader的猜想为\（K = 2 \）是类分叉准单峰毛虫，其包括每毛虫和顺序每一棵树真米直径至少\（M-4 \） 。接下来，我们不考虑树的类别，而考虑周长条件为2的连通图。然后，我们证明对于每个2连通图，Mader猜想都是正确的ģ与\（克（G）\ GE \增量（G）-8- \） ，其中克（ģ）和\（\增量（G）\）表示的周长ģ和在顶点的最小程度ģ，分别。此外，我们显示，对于每2-连通图G ^与\（克（G）\ GE \增量（G）-7 \） ，下界的\（M + 2 \）上\（\增量（G）如果\（m \ ge 10 \），则可以将Mader猜想中的\）改进为\（m + 1 \）。此外，g（G上的\（\ delta（G）-8 \）（分别是\（\ delta（G）-7 \））的下界）如果没有六个（分别是四个）可以提高到\（\ delta（G）-9 \）（分别是\（\ delta（G）-8 \）和\（m \ ge 11 \）） ）的长度周期克（ģ）具有长度的共同路径\（\左\ lceil \压裂{克（G）} {2} \右\ rceil -1 \）中g ^。我们还证明了Mader猜想对于每个带有\（g ^ \ circ（G）\ ge \ delta（G）-8 \）的2个连通图G都成立，其中\（g ^ \ circ（G）\）是G的重叠周长。Mader的猜想不仅在理论上而且在实践上都很有趣，因为它可以应用于通信网络中的容错问题。我们的证明导致\（O（| V（G）| ^ 4）\）时间算法可用于在满足假设的给定2连通图G中找到所需的子树。