An algorithm with a delay of $\mathcal{O}(k\mathrm{\Delta })$ for enumerating connected induced subgraphs of size $k$

The problem of enumerating connected subgraphs of a given size in a graph has been extensively studied in recent years. In this paper, we propose an algorithm with a delay of $O(k\mathrm{\Delta })$ for enumerating all connected induced subgraphs of size $k$ in an undirected graph $G=(V,E)$, where $k$ and $\mathrm{\Delta }$ are respectively the size of subgraphs and the maximum degree of $G$. The algorithm requires a preprocessing step of $O(\mathrm{\mid }V\mathrm{\mid }+\mathrm{\mid }E\mathrm{\mid })$ time to compute a depth-first search traversal order. The proposed algorithm improves upon the current best delay bound $O(k^2\mathrm{\Delta })$ for the connected induced subgraph enumeration problem in the literature.