The problem of enumerating all connected induced subgraphs of a given order $k$ from a given graph arises in many practical applications: bioinformatics, information retrieval, processor design,to name a few. The upper bound on the number of connected induced subgraphs of order $k$ is $n\cdot\frac{(e\Delta)^{k}}{(\Delta-1)k}$, where $\Delta$ is the maximum degree in the input graph $G$ and $n$ is the number of vertices in $G$. In this short communication, we first introduce a new neighborhood operator that is the key to design reverse search algorithms for enumerating all connected induced subgraphs of order $k$. Based on the proposed neighborhood operator, three algorithms with delay of $O(k\cdot min\{(n-k),k\Delta\}\cdot(k\log{\Delta}+\log{n}))$, $O(k\cdot min\{(n-k),k\Delta\}\cdot n)$ and $O(k^2\cdot min\{(n-k),k\Delta\}\cdot min\{k,\Delta\})$ respectively are proposed. The first two algorithms require exponential space to improve upon the current best delay bound $O(k^2\Delta)$\cite{4} for this problem in the case $k>\frac{n\log{\Delta}-\log{n}-\Delta+\sqrt{n\log{n}\log{\Delta}}}{\log{\Delta}}$ and $k>\frac{n^2}{n+\Delta}$ respectively.
翻译:从给定的图表中列出所有连接的 $k{{ 问题 { 从给定的图表 { ${ { $ 的子图问题在许多实际应用中产生 : 生物信息、 信息检索、 处理器设计, 来命名几个。 连接的 源子子图数的上限是 $n\ cd\\ frac{ (e\ Delta)\\\\ k} k}, 其中$\ Delta} (k\ log$_ G$ 和 $ 美元) 输入图中的最大度是 $G$( $$$ ) 的顶点 。 在此简短的通信中, 我们首先引入一个新的邻居操作器, 这是设计反向搜索算出所有连接的 $k$k$k$ 的子图 。 根据拟议的邻居操作员, 三种算法, $( k\\\\\\\\\\\\\\\\\\\\\ k, k) k, k\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\</s>