| Summary: | Considering a fuzzy graph G is simple and can be connected and considered as a subset <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>=</mo><mfenced open="{" close="}" separators="|"><mrow><mfenced separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>σ</mi><mfenced separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mo>,</mo><mfenced separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>σ</mi><mfenced separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfenced></mrow></mfenced><mo>,</mo><mo>…</mo><mfenced separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><mi>σ</mi><mfenced separators="|"><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></mfenced></mrow></mfenced></mrow></mfenced><mo>,</mo><mo> </mo><mo>|</mo><mi>H</mi><mo>|</mo><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>; then, every two pairs of elements of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>−</mo><mi>H</mi></mrow></semantics></math></inline-formula> have a unique depiction with the relation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi></mrow></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi></mrow></semantics></math></inline-formula> can be termed as a fuzzy resolving set (FRS). The minimal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi></mrow></semantics></math></inline-formula> cardinality is regarded as the fuzzy resolving number (FRN), and it is signified by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mi>r</mi><mfenced separators="|"><mrow><mi>G</mi></mrow></mfenced></mrow></semantics></math></inline-formula>. An independence set is discussed on the FRS, fuzzy resolving domination set (FRDS), and Fuzzy modified antimagic resolving set (FMARS). In this paper, we discuss the independency of FRS and FMARS in which an application has also been developed.
|
|---|