IEEE Access (Jan 2020)
On Mixed Metric Dimension of Some Path Related Graphs
Abstract
A vertex $k\in V_{G}$ determined two elements (vertices or edges) $\ell,m \in V_{G}\cup E_{G}$ , if $d_{G}(k,\ell)\neq d_{G}(k,m)$ . A set $R_ {\text {m}}$ of vertices in a graph $G$ is a mixed metric generator for $G$ , if two distinct elements (vertices or edges) are determined by some vertex set of $R_ {\text {m}}$ . The least number of elements in the vertex set of $R_ {\text {m}}$ is known as mixed metric dimension, and denoted as $dim_{m}(G)$ . In this article, the mixed metric dimension of some path related graphs is obtained. Those path related graphs are $P^{2}_{n}$ the square of a path, $T(P_{n})$ total graph of a path, the middle graph of a path $M(P_{n})$ , and splitting graph of a path $S(P_{n})$ . We proved that these families of graphs have constant and unbounded mixed metric dimension, respectively. We further presented an improved result for the metric dimension of the splitting graph of a path $S(P_{n})$ .
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