The k-metric dimension of a graph

As a generalization of the concept of a metric basis, this article introduces the notion of $k$-metric basis in graphs. Given a connected graph $G=(V,E)$, a set $S\subseteq V$ is said to be a $k$-metric generator for $G$ if the elements of any pair of different vertices of $G$ are distinguished by at least $k$ elements of $S$, i.e., for any two different vertices $u,v\in V$, there exist at least $k$ vertices $w_1,w_2,...,w_k\in S$ such that $d_G(u,w_i)\ne d_G(v,w_i)$ for every $i\in \{1,...,k\}$. A metric generator of minimum cardinality is called a $k$-metric basis and its cardinality the $k$-metric dimension of $G$. A connected graph $G$ is $k$-metric dimensional if $k$ is the largest integer such that there exists a $k$-metric basis for $G$. We give a necessary and sufficient condition for a graph to be $k$-metric dimensional and we obtain several results on the $k$-metric dimension.

[1]  Glenn G. Chappell,et al.  Bounds on the metric and partition dimensions of a graph , 2008, Ars Comb..

[2]  Ortrud R. Oellermann,et al.  The strong metric dimension of graphs and digraphs , 2007, Discret. Appl. Math..

[3]  Ping Zhang,et al.  Conditional resolvability in graphs: a survey , 2004, Int. J. Math. Math. Sci..

[4]  András Sebö,et al.  On Metric Generators of Graphs , 2004, Math. Oper. Res..

[5]  Juan A. Rodríguez-Velázquez,et al.  A note on the partition dimension of Cartesian product graphs , 2010, Appl. Math. Comput..

[6]  Azriel Rosenfeld,et al.  Landmarks in Graphs , 1996, Discret. Appl. Math..

[7]  Ioan Tomescu Discrepancies between metric dimension and partition dimension of a connected graph , 2008, Discret. Math..

[8]  Michael A. Henning,et al.  Locating and total dominating sets in trees , 2006, Discret. Appl. Math..

[9]  N. Duncan Leaves on trees , 2014 .

[10]  Ronald D. Dutton,et al.  Resolving domination in graphs , 2003 .

[11]  Juan A. Rodríguez-Velázquez,et al.  On the metric dimension of corona product graphs , 2011, Comput. Math. Appl..

[12]  P. Cameron,et al.  Base size, metric dimension and other invariants of groups and graphs , 2011 .

[13]  Mark E. Johnson Browsable structure-activity datasets , 1999 .

[14]  Gary Chartrand,et al.  Resolvability and the upper dimension of graphs , 2000 .

[15]  Gary Chartrand,et al.  Resolvability in graphs and the metric dimension of a graph , 2000, Discret. Appl. Math..

[16]  Juan A. Rodríguez-Velázquez,et al.  On the strong metric dimension of corona product graphs and join graphs , 2012, Discret. Appl. Math..

[17]  M. Johnson,et al.  Structure-activity maps for visualizing the graph variables arising in drug design. , 1993, Journal of biopharmaceutical statistics.

[18]  Ortrud R. Oellermann,et al.  The partition dimension of Cayley digraphs , 2006 .

[19]  Gary Chartrand,et al.  The partition dimension of a graph , 2000 .

[20]  Ping Zhang,et al.  The local metric dimension of a graph , 2010 .

[21]  David R. Wood,et al.  On the Metric Dimension of Cartesian Products of Graphs , 2005, SIAM J. Discret. Math..