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⋅(eΔ)k(Δ−1)k, where Δ 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⋅min{(n−k),kΔ}⋅(klog⁡Δ+log⁡n)), O(k⋅min{(n−k),kΔ}⋅n) and O(k2⋅min{(n−k),kΔ}⋅min{k,Δ}) respectively are proposed. The first two algorithms require exponential space to improve upon the current best delay bound O(k2Δ)[4] for this problem in the case k>nlog⁡Δ−log⁡n−Δ+nlog⁡nlog⁡Δlog⁡Δ and k>n2n+Δ respectively.