M. Kamal Kumar and R. Murali

Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132

Volume 21, 2015, Number 2, Pages 80–88

**Full paper (PDF, 144 Kb)**

## Details

### Authors and affiliations

M. Kamal Kumar

*Department of the Mathematics, CMR Institute of Technology
Bangalore 560037, India
*

R. Murali

*Department of the Mathematics, Dr. Ambedkar Institute of Technology
Bangalore, India*

### Abstract

A subset *S* of the vertex set of a graph *G* is called a dominating set of *G* if each vertex of *G*is either in *S* or adjacent to at least one vertex in *S*. A partition *D* = {*D*1, *D*2, …, *Dk*} of the vertex set of *G* is said to be a domatic partition or simply a *d*-partition of *G* if each class *Di*of *D* is a dominating set in *G*. The maximum cardinality taken over all *d*-partitions of *G* is called the domatic number of *G* denoted by *d*(*G*). A graph *G* is said to be domatically critical or *d*-critical if for every edge *x* in *G*, *d*(*G* – *x*) < *d*(*G*), otherwise *G* is said to be domatically non *d*-critical. The embedding index of a non *d*-critical graph *G* is defined to be the smallest order of a *d*-critical graph *H* containing *G* as an induced subgraph denoted by *θ*(*G*) . In this paper, we find the *θ*(*G*) for the Barbell graph, the Lollipop graph and the Tadpole graph.

### Keywords

- Domination number
- Domatic partition
- Domatic number
- d-Critical graphs

### AMS Classification

- 05C69

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## Related papers

## Cite this paper

Kamal Kumar, M., & R. Murali (2015). Embedding index in some classes of graphs. *Notes on Number Theory and Discrete Mathematics*, 21(2), 80-88.