### Abstract

Let G = (V, E) be a graph and P = {V_{1}, V_{2}, ..., V_{k}} be a partition of V of order k ≥ 1. For each set V_{r} in P, remove the edges of G inside V_{r} and add the edges Ḡ, (the complement of G) joining the vertices V_{r}. The graph G^{P}_{k} (i) thus obtained is called the k(i)-complement of G with respect to P. The graph G is k(i)-self complementary (k(i)-s.c) if G^{P}_{k} (i) ≅ G for some partition P of V of order k. Further, G is k(i)co-self complementary (k(i)-co-s.c.) if G^{P}_{k}(i) ≅ Ḡ. We determine (1) all k(i)-s.c trees for k = 2, 3, and (2) 2(i)-s.c. unicyclic graphs. Also, some necessary conditions for a tree/unicyclic graph to be k(i)-s.c. are obtained. We indicate how to obtain characterizations of all k(i)-co.s.c. trees, unicyclic graphs and forests from known results.

Original language | English |
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Pages (from-to) | 625-639 |

Number of pages | 15 |

Journal | Indian Journal of Pure and Applied Mathematics |

Volume | 29 |

Issue number | 6 |

Publication status | Published - 01-06-1998 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Indian Journal of Pure and Applied Mathematics*,

*29*(6), 625-639.