### Abstract

Let k and n be positive integers. Let G be a graph of order n≤4k, and let n = ∑^{k}_{i=1} a_{i} be a partition of n into k positive integers a_{i} with 1≤a_{i}≤4. In Matsunaga (Preprint, 1995) one of the authors proved that if G is 2-connected and the minimum degree of G is at least k, then with certain types of exceptions being allowed, the vertex set of G can be decomposed into k disjoint subsets A_{l},...,A_{k} so that |A_{i}| = a_{i} and "the subgraph induced by A_{i} is connected" for all i, 1≤i≤k. In the present paper we extend this result for connected graphs in general. The paper also features the existence of D3-paths with a prescribed endvertex in a graph. (A D3-path in a graph is a path of the graph whose removal leaves no component of order at least 3.)

Original language | English |
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Pages (from-to) | 87-104 |

Number of pages | 18 |

Journal | Discrete Mathematics |

Volume | 213 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - 2000 Feb 28 |

Event | Selected Topics in Discrete Mathematics - Warsaw, Poland Duration: 1996 Aug 26 → 1996 Sep 28 |

### Keywords

- D-path
- Graph decomposition

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

_{3}-paths with a prescribed endvertex.

*Discrete Mathematics*,

*213*(1-3), 87-104. https://doi.org/10.1016/S0012-365X(99)00170-3