### 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 |
---|---|

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.

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*Indian Journal of Pure and Applied Mathematics*, vol. 29, no. 6, pp. 625-639.

**Generalized complements of a graph.** / Sampathkumar, E.; Pushpa Latha, L.; Venkatachalam, C. V.; Bhat, Pradeep.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Generalized complements of a graph

AU - Sampathkumar, E.

AU - Pushpa Latha, L.

AU - Venkatachalam, C. V.

AU - Bhat, Pradeep

PY - 1998/6/1

Y1 - 1998/6/1

N2 - Let G = (V, E) be a graph and P = {V1, V2, ..., Vk} be a partition of V of order k ≥ 1. For each set Vr in P, remove the edges of G inside Vr and add the edges Ḡ, (the complement of G) joining the vertices Vr. The graph GPk (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 GPk (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 GPk(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.

AB - Let G = (V, E) be a graph and P = {V1, V2, ..., Vk} be a partition of V of order k ≥ 1. For each set Vr in P, remove the edges of G inside Vr and add the edges Ḡ, (the complement of G) joining the vertices Vr. The graph GPk (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 GPk (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 GPk(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.

UR - http://www.scopus.com/inward/record.url?scp=0032105171&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032105171&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032105171

VL - 29

SP - 625

EP - 639

JO - Indian Journal of Pure and Applied Mathematics

JF - Indian Journal of Pure and Applied Mathematics

SN - 0019-5588

IS - 6

ER -