| Summary: | An Italian dominating function of <i>G</i> is a function <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>:</mo> <mi>V</mi> <mo>(</mo> <mi>G</mi> <mo>)</mo> <mo>→</mo> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>}</mo> </mrow> </semantics> </math> </inline-formula>, for every vertex <i>v</i> such that <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>v</mi> <mo>)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, it holds that <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mo>∑</mo> <mrow> <mi>u</mi> <mo>∈</mo> <mi>N</mi> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </msub> <mi>f</mi> <mrow> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> <mo>≥</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>. The Italian domination number <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>γ</mi> <mi>I</mi> </msub> <mrow> <mo>(</mo> <mi>G</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is the minimum weight of an Italian dominating function on <i>G</i>. In this paper, we determine the exact values of the Italian domination numbers of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>n</mi> <mo>,</mo> <mn>3</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>.
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