### Abstract

Let X be a subset of vertices of an undirected graph G = (V, E). G is X-critical if it is indecomposable and its induced subgraph on X vertices is also indecomposable, but all induced subgraphs on V - {w} are decomposable for all w ∈ V - X. We present two results in this paper. The first result states that if G is X-critical, then for every w ∈ V - {x}, G[V -{w}] has a unique non-trivial module and its cardinality is either 2 or |V| - 2. The second result is that the vertices of V - X can be paired up as (a_{1}, b_{1}), . . . , (a_{k}, b_{k}) such that induced subgraphs on subset V - {a_{j1}, b_{j1}, . . . , a_{js}, b_{js}} are also X-critical for any collection of pairs {(a _{j1}, b_{j1}), . . . ,(a_{js}, b_{js})}.

Original language | English (US) |
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Pages (from-to) | 690-700 |

Number of pages | 11 |

Journal | Lecture Notes in Computer Science |

Volume | 3595 |

DOIs | |

State | Published - Jan 1 2005 |

Event | 11th Annual International Conference on Computing and Combinatorics, COCOON 2005 - Kunming, China Duration: Aug 16 2005 → Aug 29 2005 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Lecture Notes in Computer Science*,

*3595*, 690-700. https://doi.org/10.1007/11533719_70