SSD Set System, Graph Decomposition and Hamiltonian Cycle
<|reference_start|>SSD Set System, Graph Decomposition and Hamiltonian Cycle: In this paper, we first study what we call Superset-Subset-Disjoint (SSD) set system. Based on properties of SSD set system, we derive the following (I) to (IV):   (I) For a nonnegative integer $k$ and a graph $G=(V,E)$ with $|V|\ge2$, let $X_1,X_2,\dots,X_q\subsetneq V$ denote all maximal proper subsets of $V$ that induce $k$-edge-connected subgraphs. Then at least one of (a) and (b) holds: (a) $\{X_1,X_2,\dots,X_q\}$ is a partition of $V$; and (b) $V\setminus X_1, V\setminus X_2,\dots,V\setminus X_q$ are pairwise disjoint.   (II) For $k=1$ and a strongly-connected digraph $G$, whether $V$ is in (a) and/or (b) can be decided in $O(n+m)$ time and we can generate all such $X_1,X_2,\dots,X_q$ in $O(n+m+|X_1|+|X_2|+\dots+|X_q|)$ time, where $n=|V|$ and $m=|E|$.   (III) For a digraph $G$, we can enumerate in linear delay all vertex subsets of $V$ that induce strongly-connected subgraphs.   (IV) A digraph is Hamiltonian if there is a spanning subgraph that is strongly-connected and in the case (a).<|reference_end|>
@article{shota2024ssd,
    title={SSD Set System, Graph Decomposition and Hamiltonian Cycle},
    author={Kan Shota, Kazuya Haraguchi},
    journal={arXiv preprint arXiv:2408.04615},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.04615},
    primaryClass={cs.DS cs.DM math.CO}
}