Computing Metric Dimension of Power of Total Graph
Sehar Nawaz, Murtaza Ali, Mushtaq Ahmad Khan, Sahib Khan · IEEE Access · 2021
For a connected graph$\mathcal {G}$, the distance$d(u, v)$between any two vertices$u$,$v~\in ~\mathcal {V(G)}$is defined as;$d(u,v)=\min _{\mathcal {P}_{u,v}}\{\text {length of}~\mathcal {P}_{uv}\}$i.e the minimum length of path connecting any two vertices. Two vertices$u$,$v$of a graph$G$are said to be resolved by a vertices$\mathcal {W}$of a graph$\mathcal {G}$if$d(u,w) eq d(v,w)$. An ordered set$\mathcal {W} = \{w_{1},w_{2},w_{3},\ldots,w_{k}\}\,\,\in \,\,V(G)$is resolving set for a connected graph$G$, if for any two vertices$u$,$v$there exists$w_{i} \in W$such that$d(u,w_{i}) eq d(v, w_{i})$. The representation of vertices$v$w.r.t$\mathcal {W}$is represented by$r(v|\mathcal {W})$which is$k$-vector($k$-tuple)$(d(v,w_{1}),d(v,w_{2}),d(v,w_{3}),\ldots,d(v,w_{k}))$. If the representation of all the vertices of graph$\mathcal {G}$are different w.r.t$\mathcal {W}$then$\mathcal {W}$is the resolving set for$G$[1]. A resolving set that contains minimum numbers of vertices ismetric basisfor$\mathcal {G}$. The cardinality of smallest resolving set is themetric dimensionof$\mathcal {G}$, represented by$dim(\mathcal {G})$[2],[3]. A family of connected graph$\mathcal {G}$has unbounded metric dimension if$dim(\mathcal {G})$depends on order of a graph. In this paper, we show that total graph of path power three and four have unbounded metric dimension. We also proved some results on edges of power of path graph.