| Summary: | In a simple connected undirected graph <i>G</i>, an ordered set <i>R</i> of vertices is called a resolving set if for every pair of distinct vertices <i>u</i> and <i>v</i>, there is a vertex <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mo>∈</mo><mi>R</mi></mrow></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>≠</mo><mi>d</mi><mo>(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow></semantics></math></inline-formula>. A resolving set <i>F</i> for the graph <i>G</i> is a <i>fault-tolerant resolving set</i> if for each <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>F</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>∖</mo><mo>{</mo><mi>v</mi><mo>}</mo></mrow></semantics></math></inline-formula> is also a resolving set for <i>G</i>. In this article, we determine an optimal fault-resolving set of <i>r</i>-th power of any path <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>P</mi><mi>n</mi></msub></semantics></math></inline-formula> when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mi>r</mi><mo>(</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>2</mn></mrow></semantics></math></inline-formula>. For the other values of <i>n</i>, we give bounds for the size of an optimal fault-resolving set. We have also presented an algorithm to construct a fault-tolerant resolving set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>P</mi><mi>m</mi><mi>r</mi></msubsup></semantics></math></inline-formula> from a fault-tolerant resolving set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>P</mi><mi>n</mi><mi>r</mi></msubsup></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo><</mo><mi>n</mi></mrow></semantics></math></inline-formula>.
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