### Abstract

Let G be a graph of positive size q, and let n be that positive integer for which (Figure presented.). It was conjectured by Alavi et al that the G can be decomposed into subgraphs G_{1}, G_{2}, …,G_{n} such that G_{i} is isomorphic to a proper subgraph of G_{i+1} for 1 ≤ i ≤ n−1. Since the conjecture was posed around 25 years many special classes of graphs have been verified to have ascending subgraph decomposition defined above. Because of the interesting nature of the structure of graphs, many researchers have devoted themselves to work on this problem and also provided several extended versions of such decompositions. Mainly, the size of G and the structure of G_{i}’s are concerned. As hard as we work on this problem, we find the fascinating inside of this seemingly easy conjecture. This is the reason, we decide to make a survey of this decomposition and hopefully, we would (be very happy to) see the ASD conjecture proved to be true in the near future by some talented researchers.

Original language | English |
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Pages (from-to) | 1135-1149 |

Number of pages | 15 |

Journal | Journal of Discrete Mathematical Sciences and Cryptography |

Volume | 20 |

Issue number | 5 |

DOIs | |

State | Published - 4 Jul 2017 |

### Keywords

- ascending subgraph decomposition of graph
- matching
- path
- star
- subset partition

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## Cite this

*Journal of Discrete Mathematical Sciences and Cryptography*,

*20*(5), 1135-1149. https://doi.org/10.1080/09720529.2015.1032619