On the metric dimension of HDN
The concept of metric basis is useful for robot navigation. In graph G, a robot is aware of its current location by sending signals to obtain the distances between itself and the landmarks in G. Its position is determined uniquely in G if it knows its distances to sufficiently many landmarks. The metric basis of G is defined as the minimum set of landmarks such that all other vertices in G can be uniquely determined and the metric dimension of G is defined as the cardinality of the minimum set of landmarks. The major contribution of this paper is that we have partly solved the open problem proposed by Manuel et al. [9], by proving that the metric dimension of HDN1(n) HDN1 ( n ) and HDN2(n) HDN2 ( n ) are either 3 or 4. However, the problem of finding the exact metric dimension of HDN networks is still open.
metric dimension metric basis hdn1(n) hdn1 ( n ) hdn1(n) hdn1 ( n ) hdn1(n) hdn1 ( n ) hdn1 ( n ) hdn2(n) hdn2 ( n ) hdn2(n) hdn2 ( n ) hdn2(n) hdn2 ( n ) hdn2 ( n )