On the local metric dimension of t-fold wheel, Pn o Km, and generalized fan

Abstract

<p>Let <span class="math"><em>G</em></span> be a connected graph and let <span class="math"><em>u</em>, <em>v</em></span> <span class="math"> ∈ </span> <span class="math"><em>V</em>(<em>G</em>)</span>. For an ordered set <span class="math"><em>W</em> = {<em>w</em><sub>1</sub>, <em>w</em><sub>2</sub>, ..., <em>w</em><sub><em>n</em></sub>}</span> of <span class="math"><em>n</em></span> distinct vertices in <span class="math"><em>G</em></span>, the representation of a vertex <span class="math"><em>v</em></span> of <span class="math"><em>G</em></span> with respect to <span class="math"><em>W</em></span> is the <span class="math"><em>n</em></span>-vector <span class="math"><em>r</em>(<em>v</em>∣<em>W</em>) = (<em>d</em>(<em>v</em>, <em>w</em><sub>1</sub>), <em>d</em>(<em>v</em>, <em>w</em><sub>2</sub>), ..., </span> <span class="math"><em>d</em>(<em>v</em>, <em>w</em><sub><em>n</em></sub>))</span>, where <span class="math"><em>d</em>(<em>v</em>, <em>w</em><sub><em>i</em></sub>)</span> is the distance between <span class="math"><em>v</em></span> and <span class="math"><em>w</em><sub><em>i</em></sub></span> for <span class="math">1 ≤ <em>i</em> ≤ <em>n</em></span>. The set <span class="math"><em>W</em></span> is a local metric set of <span class="math"><em>G</em></span> if <span class="math"><em>r</em>(<em>u</em> ∣ <em>W</em>) ≠ <em>r</em>(<em>v</em> ∣ <em>W</em>)</span> for every pair <span class="math"><em>u</em>, <em>v</em></span> of adjacent vertices of <span class="math"><em>G</em></span>. The local metric set of <span class="math"><em>G</em></span> with minimum cardinality is called a local metric basis for <span class="math"><em>G</em></span> and its cardinality is called a local metric dimension, denoted by <span class="math"><em>l</em><em>m</em><em>d</em>(<em>G</em>)</span>. In this paper we determine the local metric dimension of a <span class="math"><em>t</em></span>-fold wheel graph, <span class="math"><em>P</em><sub><em>n</em></sub></span> <span class="math"> ⊙ </span> <span class="math"><em>K</em><sub><em>m</em></sub></span> graph, and generalized fan graph.</p>