On the Fault Tolerant Partition Resolvability of Toeplitz Networks

Abstract

In any interconnection network, fault tolerance is the most desirable property to achieve reliability. Toeplitz networks are used as interconnection networks due their smaller diameter, symmetry, simpler routing, high connectivity, and reliability. The partition dimension of a network is presented as an extension of metric dimension of networks. Its applications can be seen in several areas including robot navigation, network designing, image processing, and chemistry. In this article, the fault tolerant partition dimension, <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <msub> <mrow> <mtext>pd</mtext> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> <mfenced open="〈" close="〉" separators="|"> <mrow> <mn>1</mn> <mo>,</mo> <mi>t</mi> </mrow> </mfenced> </mrow> </mfenced> </math> , of Toeplitz networks, is shown to be bounded below by 4 for <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>t</mi> <mo>≥</mo> <mn>2</mn> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </math> , whereas it is bounded above by 5 for <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mi>t</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>14</mn> </math> . Further, it is shown that the exact value of <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <msub> <mrow> <mtext>pd</mtext> </mrow> <mrow> <mn>2</mn> </mrow> </msub> <mfenced open="(" close=")" separators="|"> <mrow> <msub> <mrow> <mi>T</mi> </mrow> <mrow> <mi>n</mi> </mrow> </msub> <mfenced open="〈" close="〉" separators="|"> <mrow> <mn>1</mn> <mo>,</mo> <mi>t</mi> </mrow> </mfenced> </mrow> </mfenced> </math> equals 4 for <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <mi>t</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </math> ; <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6"> <mi>t</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mfenced open="{" close="}" separators="|"> <mrow> <mn>5,6</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>13</mn> </mrow> </mfenced> </math> ; and <math xmlns="http://www.w3.org/1998/Math/MathML" id="M7"> <mi>t</mi> <mo>≥</mo> <mn>4</mn> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mfenced open="{" close="}" separators="|"> <mrow> <mi>t</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mi>t</mi> <mo>+</mo> <mn>3</mn> <mo>,</mo> <mi>t</mi> <mo>+</mo> <mn>4</mn> </mrow> </mfenced> </math> .