### The geodetic domination number of comb product graphs

#### Abstract

A subset *S* of vertices in graph *G* is called a *geodetic set* if every vertex in *V*(*G*) \ *S* lies on a shortest path between two vertices in *S*. A subset *S* of vertices in *G* is called a *dominating set* if every vertex in *V*(*G*) \ *S* is adjacent to a vertex in *S*. The set *S* is called a *geodetic dominating set* if *S* is both geodetic and dominating sets. The *geodetic domination number* of *G*, denoted by *γ _{g}*(

*G*), is the minimum cardinality of geodetic domination sets in

*G*. The

*comb product*of connected graphs

*G*and

*H*at vertex o ∈

*V*(

*H*), denoted by

*G ∇*, is a graph obtained by taking one copy of

_{o}H*G*and |

*V*(

*G*)| copies of

*H*and identifying the

*i*

^{th}copy of

*H*at the vertex

*o*to the

*i*

^{th}vertex of

*G*. In this paper, we determine an exact value of

*γ*(

_{g}*G ∇*) for any connected graphs

_{o}H*G*and

*H*.

#### Keywords

#### Full Text:

PDFDOI: http://dx.doi.org/10.5614/ejgta.2020.8.2.13

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