# The Application of Fault-Tolerant Partition Resolvability in Cycle-Related Graphs
## 1 Introduction and Preliminaries
[PERSON] et al. discussed the FTMD of circulant graphs (see [14]). [PERSON] et al. studied this graph parameter on certain inter-connection networks (see [15]). For a given \(\lambda\)-ordered partition of vertices of a connected graph \(\mathrm{Y}\), represented as \(\aleph=\{\aleph_{1},\aleph_{2},\ldots,\aleph_{\lambda}\}\), the representation (distance code) of the vertex \(\tau\in V(\mathrm{Y})\) is the vector \(r(\tau|\aleph)=(d(\tau,\aleph_{1}),d(\tau,\aleph_{2}),\ldots,d(\tau,\aleph_{ \lambda}))\). The partition \(\aleph\) is called a resolving partition of \(\mathrm{Y}\) if all the vertices of \(\mathrm{Y}\) have unique representations (distance codes) with respect to \(\aleph\). The minimum number of sets in the resolving partition for \(\mathrm{Y}\) is known as the partition dimension (PD) of \(\mathrm{Y}\) and is represented by \(pd(\mathrm{Y})\). The \(pd(\mathrm{Y})\) of various classes of connected graphs could be constant, bounded or unbounded. [PERSON] et al. characterized the graphs with \(pd(\mathrm{Y})\) 2 or \(\alpha\) (see [4]). Sharp bounds for the PD of convex polytopes and flower graphs were provided by [PERSON] et al. (see [16]). The PD of kayak paddle and cycle with chord graphs was discussed by [PERSON] et al. (see [17]). [PERSON] et al. discussed the PD of honey comb, hexagonal cage and quartz networks (see [18]). [PERSON] et al. provided the bounds on the PD of families of convex polytopes with pendent edges (see [19]). The PD of a thorn of fan graph and certain classes of series parallel graphs were computed by [PERSON] and [PERSON] et al., respectively (see [20; 21]). The complexity in computing the \(\beta(\mathrm{Y})\) of general graphs was investigated by [PERSON] et al. and [PERSON] et al. (see [22; 23]). As the PD problem is a generalized variant of a metric dimension, therefore, \(pd(\mathrm{Y})\) is also an NP-hard problem. The advancement of the concept of the \(pd(\mathrm{Y})\) of a graph as a fault-tolerant partition dimension (FTPD) of graphs was proposed by [PERSON] et al. (see [24]). For a given \(\lambda\)-ordered partition of vertices of a connected graph \(\mathrm{Y}\), represented by \(\aleph=\{\aleph_{1},\aleph_{2},\ldots,\aleph_{\lambda}\}\), the partition \(\aleph\) of \(\mathrm{Y}\) is called a fault-tolerant resolving partition if every pair of distinct vertices \(\rho,\tau\in V(\mathrm{Y})\), \(r(\rho|\aleph)\) and \(r(\tau|\aleph)\) has representations (distance codes) which differ by at least two places. The FTPD of \(\mathrm{Y}\) is defined as \(\min\{|\aleph|:\aleph\text{ is fault-tolerant resolving partition of }\mathrm{Y}\}\) and is denoted by \(\mathcal{P}(\mathrm{Y})\). The FTPDs of certain classes of graphs, namely homogeneous caterpillar (see [25]), cyclic network (see [26]) and tadpole and necklace graphs (see [27]), are discussed by [PERSON] et al. The FTPDs of toeplitz networks (see [28]), convex polytopes (see [29]) and circulant graphs having a \(\{1,2\}\) connection set (see [30]) were discussed by [PERSON] et al. For further studies on FTPDs, we refer to [31; 32]. In this paper, we protract these studies by computing the FTPD of a kayak paddle graph and flower graph. [PERSON] et al. revealed the following results on \(pd(\mathrm{Y})^{\prime}\)  
**Proposition 1** ([4]). : _Let \(Y\) be a graph. Then, the following hold:_ \begin{tabular}{l l} _(a)_ & \(pd(Y)\leq\beta(Y)+1\)_;_ \\ _(b)_ & \(pd(Y)=2\) _iff_ \(Y=P_{\alpha}\)_, where_ \(P_{\alpha}\) _is a path._ \\ \end{tabular}   
[PERSON] et al. introduced successive results for \(\mathcal{P}(Y)\):  
**Proposition 2** ([24]). : _Let \(Y\) be a graph. Then, the following hold:_ \begin{tabular}{l l} _(a)_ & _For_ \(\alpha\geq 3\)_,_ \(3\leq\mathcal{P}(Y)\leq\alpha\)_;_ \\ _(b)_ & _For_ \(\alpha\geq 2\)_,_ \(\mathcal{P}(Y)\leq\beta^{{}^{\prime}}(Y)+1\)_._ \\ \end{tabular}   
In 2022, [PERSON] et al. provided an important result for the FTPD of a graph in which the degree of any vertex was at least four. This result is given in the following lemma:  
**Lemma 1** ([29]).