Algorithms with improved delay for enumerating connected induced subgraphs of a large cardinality

Abstract

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.