## AbstractA subgraph $H$ of an acyclic digraph $D$ is convex if there is no directed path between vertices of $H$ which contains an arc not in $H$. A digraph $D$ is connected if the underlying undirected graph of $D$ is connected. We construct an algorithm for enumeration of all connected convex subgraphs of a connected acyclic digraph $D$ of order $n$. The time complexity of the algorithm is $O(n\cdot cc(D))$, where $cc(D)$ is the number of connected convex subgraphs in $D$. The space complexity is $O(n^2).$ Connected convex subgraphs of connected acyclic digraphs are of interest in the area of modern embedded processors technology. Our computational experiments %% 'slightly' removed demonstrate that our algorithm is better than the state-of-the-art algorithm of Chen, Maskell and Sun. Moreover, unlike the algorithm of Chen, Maskell and Sun, our algorithm has a provable (almost) optimal worst time complexity. Using the same approach, we design an algorithm for generating all connected induced subgraphs of a connected undirected graph $G$. The complexity of the algorithm is $O(n\cdot c(G)),$ where $n$ is the order of $G$ and $c(G)$ is the number of connected induced subgraphs of $G.$ The previously reported algorithm for connected induced subgraph enumeration is of running time $O(mn\cdot c(G))$, where $m$ is the number of edges in $G.$
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