THE GEODETIC FAULT TOLERANT DOMINATION NUMBER OF A GRAPH

For a connected graph $G=(V,E)$, a set $F\subseteq V$ of vertices in $G$ is called dominating set if every vertex not in $F$ has at least one neighbor in $F$. A dominating set $F\subseteq V$ is called fault tolerant dominating set if $F-\{v\}$ is dominating set for every $v\in F$. A fault tolerant dominating set is said to be geodetic fault tolerant dominating set if $I[F]=V$. The minimum cardinality of a geodetic fault tolerant dominating set is called geodetic fault tolerant domination number and is denoted by $\gamma_{gft}(G)$. The minimum geodetic fault tolerant dominating set is denoted by $\gamma_{gft}$-set. The geodetic fault tolerant domination number of certain classes of graphs are determined. Some general properties satisfied by this concept are studied. It is shown that for every positive integer $2< a \leq b$ there is a connected graph $G$ such that $\gamma(G)=a$, $\gamma_{g}(G)=b$ and $\gamma_{gft}(G)=a+b-2$, where $\gamma(G)$ and $\gamma_{g}(G)$ are the domination number and geodetic domination number of $G$ respectively