### On the Steiner antipodal number of graphs

#### Abstract

The Steiner *n*-antipodal graph of a graph *G* on *p* vertices, denoted by *S**A*_{n}(*G*), has the same vertex set as *G* and any *n*(2 ≤ *n* ≤ *p*) vertices are mutually adjacent in *S**A*_{n}(*G*) if and only if they are *n*-antipodal in *G*. When *G* is disconnected, any *n* vertices are mutually adjacent in *S**A*_{n}(*G*) if not all of them are in the same component. *S**A*_{n}(*G*) coincides with the antipodal graph *A*(*G*) when *n* = 2. The least positive integer *n* such that *S**A*_{n}(*G*) ≅ *H*, for a pair of graphs *G* and *H* on *p* vertices, is called the Steiner *A*-completion number of *G* over *H*. When *H* = *K*_{p}, the Steiner *A*-completion number of *G* over *H* is called the Steiner antipodal number of *G*. In this article, we obtain the Steiner antipodal number of some families of graphs and for any tree. For every positive integer *k*, there exists a tree having Steiner antipodal number *k* and there exists a unicyclic graph having Steiner antipodal number *k*. Also we show that the notion of the Steiner antipodal number of graphs is independent of the Steiner radial number, the domination number and the chromatic number of graphs.

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2019.7.2.3

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ISSN: 2338-2287

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