| Summary: | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>e</mi><mo>)</mo></mrow></semantics></math></inline-formula> be a class of numerical semigroups with multiplicity <i>m</i> and embedding dimension <i>e</i>. We call a graph <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mi>S</mi></msub></semantics></math></inline-formula> an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>e</mi><mo>)</mo></mrow></semantics></math></inline-formula>-graph if there exists a numerical semigroup <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>S</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>e</mi><mo>)</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mrow><mo>(</mo><msub><mi>G</mi><mi>S</mi></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo stretchy="false">{</mo><mi>x</mi><mo>:</mo><mi>x</mi><mo>∈</mo><mi>g</mi><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow><mo stretchy="false">}</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>E</mi><mrow><mo>(</mo><msub><mi>G</mi><mi>S</mi></msub><mo>)</mo></mrow><mo>=</mo><mrow><mo stretchy="false">{</mo><mi>x</mi><mi>y</mi><mo>⇔</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>∈</mo><mi>S</mi><mo stretchy="false">}</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>(</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> denotes the gap set of <i>S</i>. The aim of this article is to discuss the planarity of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>e</mi><mo>)</mo></mrow></semantics></math></inline-formula>-graphs for some cases where S is an irreducible numerical semigroup.
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