On multiset dimension of cylindrical graphs

Let <span class="math inline"><em>G</em> = (<em>V</em>, <em>E</em>)</span> be a simple connected graph and <span class="math inline"><em>W</em> ⊆ <em>V</em>.</span> For <span class="math inline"><em>v</em> ∈ <em>V</em>,</span> the representation multiset or m-code of <span class="math inline"><em>v</em></span> is the multiset <span class="math inline"><em>r</em><sub><em>m</em></sub>(<em>v</em>) = {<em>d</em>(<em>v</em>, <em>w</em>) ∣ <em>w</em> ∈ <em>W</em>}</span>. If no two vertices in <span class="math inline"><em>G</em></span> have equal m-codes, then <span class="math inline"><em>W</em></span> is called an m-resolving set of <span class="math inline"><em>G</em></span>. The multiset dimension <span class="math inline">md(<em>G</em>)</span> of <span class="math inline"><em>G</em></span> is the minimum possible cardinality of an m-resolving set of <span class="math inline"><em>G</em></span>, if such a set exists. If <span class="math inline"><em>G</em></span> does not possess an m-resolving set, then we say that <span class="math inline"><em>G</em></span> has infinite multiset dimension. In this paper, we show that all cylindrical graphs <span class="math inline"><em>P</em><sub><em>m</em></sub> ▫ <em>C</em><sub><em>n</em></sub></span>, where <span class="math inline"><em>m</em>, <em>n</em> ≥ 3</span>, have finite multiset dimension. In particular, we show that md<span class="math inline">(<em>P</em><sub><em>m</em></sub> ▫ <em>C</em><sub><em>n</em></sub>) ≤ 4</span> if <span class="math inline"><em>m</em> ≥ 6</span> and <span class="math inline"><em>n</em> ≥ 3</span>, or if <span class="math inline"><em>m</em> ≥ 3</span> and <span class="math inline"><em>n</em> ≥ 12</span>. Moreover, if <span class="math inline"><em>m</em> ≥ 3</span> and <span class="math inline"><em>n</em> ≥ 8<em>m</em> + 1</span>, we show that <span class="math inline"><em>P</em><sub><em>m</em></sub> ▫ <em>C</em><sub><em>n</em></sub></span> has multiset dimension 3.