Breast cancer chemical structures and their partition resolvability

1.   Introduction

Trillions of cells make up the human body. Cell division occurs naturally in all living things. Cancer develops when uncontrollable cell division occurs and spreads to neighboring tissues, forming tumors. Cancer can occur in any region of the body. Patient recovery rates have significantly improved in recent years. Cancer develops as a result of hormonal or genetic alterations in the DNA. Cancer affects people of all ages, from babies to the elderly, although it most commonly affects adults. A biopsy is essential to confirm the diagnosis if another growth, lump, or tumor appears in the body. Tumors may be cancerous or benign. Benign non-cancerous tumors do not spread to neighboring tissues. Certain benign tumors, on the other hand, can be lethal if they form in the brain.

The chance of acquiring cancer can be reduced by various variables, including keeping a healthy lifestyle, avoiding cancer-causing foods, and receiving cancer-prevention vaccinations. Tobacco use, carcinogen exposure, and cooking with Teflon-coated utensils are all carcinogens with the potential to produce the worst illness ^[1,2,3].

Cancer affects all humans, regardless of gender. Women are disproportionately impacted by breast and cervical cancer. According to 2020, million women worldwide were affected by breast cancer, with 685,000 losing their struggle against the worst illness. It is most commonly seen in the lining of milk ducts and the lobules that provide milk to these ducts. There are more than 18 types of breast cancer. Mammograms are used to detect breast cancer at an early stage. Clinical trials, immunological therapy, hormone therapy, targeted therapy, surgery, chemotherapy, and radiation therapy are all part of the treatment ^[4,5].

Assume ${G}(V ({G}), E ({G}))$ is an undirected graph of a chemical structure (network), with $V ({G})$ representing the set of primary nodes (vertex set) and $E ({G})$ representing the set of branches (edge set). The distance between two primary nodes ${v}_1, {v}_2 \in V ({G}),$ abbreviated as $d ({v}_1, {v}_2),$ is the least number of edges between the ${v}_1, {v}_2$ route. Assume $R \subset V ({G})$ is the subset of principal nodes defined by $R = \{{v}_1, {v}_2, \dots, {v}_s\},$ and consider a principal node ${v} \in V ({G}).$ A primary node's identification or position $r({v}|R)$ with regard to $R$ is really a distance $(d({v}, {v}_1), d({v}, {v}_2), \dots, d({v}, {v}_s)).$ If each primary node in $V({G})$ has a unique identity according to the ordered subset R, then this subset is termed a network resolving set. The metric of dimension is the minimal number of elements in the subset $R,$ which is indicated by the word dim (${G}$). Metric dimension is used in various applications, including combinatorial optimization, robot roving ^[6], complicated games, image processing, pharmaceutical chemistry, polymer production ^[7], and the electric field. These applications may all be found in ^{[8,9,10,11,12,13,14,15,16]}.

The Authors examined windmill graphs in terms of metric generalization in their research work ^[17,18]. The authors explored the extended version of the measured dimension graph and characterized this parameter as a function of two variables. Researchers derived metrics and upper limits on various generalized families of graphs in ^[19]. Polycyclic hydrocarbons are discussed in depth in this ^[20], along with the notion of metric and its extensions. In ^[21], symmetric graphs are created using the rooted product, and metrics and their extensions are investigated. Researchers present the concept of the hollow coronoid on the metric dimension and its generalization ^[22]. The authors investigated and measured the resolvability of quartz structure and precisely calculated metric parameters for quartz structure without considering the pendant nodes beyond the circle.

The research of ^[23] provides information on rough graphs on the themes of metric dimensions and their generalized parameters. The authors of ^[24] studied hereditary bipartite networks and how to compute the metric basis of this extended class of complex networks. The concept of pseudo-valuation on KU-algebras was examined in the research work given in ^[25], as well as the link between pseudo-valuations and KU-algebras and their generalizations. More current publications on chemical networks ^{[26,27,28,29,30]}, and the metric parameters of various chemical structures and networks may be found in ^{[31,32,33,34,35,36]}. In ^[37,38], authors discussed some modeling and algorithms for that modeling regarding some disease and their cures.

Definition 1.1. Let $R_p\subseteq V({G})$ is the s-elements proper set and $r({v}|R_p) = {d({v}|R_{p_1}), d({v}|R_{p_2}), \dots, d({v}|R_{p_s})},$ is the s-tuple distance identification of a principal node ${v}$ in association with $R_p.$ If the entire set of principal nodes have unique identifications, then $R_p$ is named as the partition resolving set of the principal node of a network ${G}.$ The least possible count of the subsets in that set of $V ({G})$ is labeled as the partition dimension $pd$ of ${G}.$

Further, the methodology is presented in Figure 1. Partition dimension is another type of dimension like metric dimension on the basics of vertices. Computing the metric dimension of different chemical structures is an NP-hard problem and adding the partition dimension is too. Because partition dimension has a more complicated structure than metric dimension, fewer precise partitions are accessible, and boundaries are frequently offered. Bounds for the partition of the generalized class of convex polytopes were reported in ^[39,40]. Presents a chemical fullerene graph, ^[41] details constraints on another chemical structure, ^[42] presents several nanotubes and sheets in the form of partition sets, and ^[43] provides the two-dimensional lattice structure.

Figure 1.  Flowchart for comouting the partition dimension of a chemical structure.

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Figure 2.  Chemical structure of Podophylb toxin drug.

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Resolvability concepts are used in various applications, including combinatorial optimization, robot roving, complicated games, image processing, pharmaceutical chemistry, polymer production, and the electric field. These applications may all be found in ^{[6,7,14,15,16]}.

2.   Main results

We will include our findings of partition locating sets of various cancer drugs structures, such as Daunorubicin, Degualin, Minocyline, Podophylb toxin, Pterocevin, and Raloxifene.

Podophylb toxin structure's order and size are $\left|{V}\left({G}{ \rm{Podophylb \;toxin}}\right)\right| = 30,$ $\left|{E}\left({G}{ \rm{Podophylb \;toxin }}\right)\right| = 33,$ respectively. The node and bond set of the Podophylb toxin drug structure is shown below. Furthermore, Figure 3 depicts the molecular graph of Podophylb toxinl and labeling employed in our findings.

Figure 3.  Graph of Podophylb toxin drug structure shown in the Figure 2.

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Figure 4.  Chemical structure of Pterocellin toxin drug.

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Theorem 2.1. Let ${G}$ be a graph of Podophylb toxin drug structure. Then the partition dimension of ${G}$ is $3.$

Proof. The graph's partition dimension of Podophylb toxin drug structure is 2. To demonstrate this point, we picked a partition locating set of cardinality 3 and stated as $R\left({G}\right) = \{R_1, R_2, R_3\},$ where $R_1 = \{{v}_{1}\},$ $R_2 = \{{v}_{25}\}$ and $R_3 = V({G}) \backslash \{{v}_{1}, {v}_{25}\}.$ To demonstrate this statement true, we have included representations of each node of the Podophylb toxin drug structure except ${v}_{1}$ and ${v}_{25}$ in Table 1.

Table 1.  Representations of the nodes w.r.t. $R$.

$r({v}_{i}|R)$	$R_{1}$	$R_{2}$	$R_{3}$	Range of ${i}$
$v_{{i} }$	${i}-1$	$8$	$0$	${i}=1, 4$
$v_{{i} }$	${i}-1$	$11-{i}$	$0$	${i}=2$
$v_{{i} }$	${i}-1$	$10-{i}$	$0$	${i}=3$
$v_{{i} }$	${i}-1$	${i}+2$	$0$	${i}=5, ..., 8$
$v_{{i} }$	${i}-1$	$19-{i}$	$0$	${i}=9$
$v_{{i} }$	$17-{i}$	$19-{i}$	$0$	${i}=10, ..., 14$
$v_{{i} }$	$17-{i}$	${i}-9$	$0$	${i}=15, 16$
$v_{{i} }$	$19-{i}$	${i}-9$	$0$	${i}=17$
$v_{{i} }$	${i}-15$	${i}-15$	$0$	${i}=19, 20$
$v_{{i} }$	${i}-15$	$25-{i}$	$0$	${i}=21, 22$
$v_{{i} }$	$29-{i}$	${i}-21$	$0$	${i}=23, 24$
$v_{{i} }$	$33-{i}$	$27-{i}$	$0$	${i}=26$
$v_{{i} }$	$35-{i}$	$32-{i}$	$0$	${i}=27$
$v_{{i} }$	$36-{i}$	$32-{i}$	$0$	${i}=28$
$v_{{i} }$	$37-{i}$	$35-{i}$	$0$	${i}=29, 30$

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Each node of graph of the Podophylb toxin medication structure's given positions is unique and meets the specifications of the partition locating set. This demonstrated that the partition locating number is correct.

Hence, proved that $pd\left({G}_{ \rm{Podophylb \;toxin}}\right)\leq3.$ To verify this statement, we must demonstrate that $pd\left({G}_{ \rm{Podophylb \;toxin}}\right)\geq3,$ and then by contradiction, we will have $pd\left({G}_{ \rm{Podophylb \;toxin}}\right) = 2.$This is not correct because this assertion is only applicable to path graphs.

Hence, proved that $pd\left({G}_{ \rm{Podophylb \;toxin}}\right) = 3.$

Pterocellin structure's order and size are $\left|{V}\left({G}_{ \rm{Pterocellin}}\right)\right| = 24,$ $\left|{E}\left({G}_{ \rm{Pterocellin }}\right)\right| = 27,$ respectively. The node and bond set of the Pterocellin drug structure are shown below. Furthermore, Figure 5 depicts the molecular graph of Pterocellin and labeling employed in our findings.

Figure 5.  Graph of Pterocellin toxin drug structure shown in the Figure 4.

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Figure 6.  Chemical structure of daunorubicin drug.

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Theorem 2.2. Let ${G}$ be a graph of Pterocellin drug structure. Then the partition locating number of ${G}$ is $3.$

Proof. The graph's partition dimension of Pterocellin drug structure is 3. To demonstrate this point, we picked a partition set of cardinality 3 and stated as $R\left({G}\right) = \{R_1, R_2, R_3\},$ where $R_1 = \{{v}_{2}\},$ $R_2 = \{{v}_{20}\}$ and $R_3 = V({G}) \backslash \{{v}_{2}, {v}_{20}\}.$ To demonstrate this statement true, we have included representations of each node of the Pterocellin drug structure except ${v}_{2}$ and ${v}_{20}$ in Table 2.

Table 2.  Representations of the vertices with respect to $R$.

$r({v}_{i}|R)$	$R_{1}$	$R_{2}$	$R_{3}$	Range of ${i}$
$v_{{i} }$	${i}$	${i}+5$	$0$	${i}=1$
$v_{{i} }$	${i}-2$	${i}+5$	$0$	${i}=2, 3$
$v_{{i} }$	${i}-2$	$10-{i}$	$0$	${i}=4, 5$
$v_{{i} }$	${i}-2$	$12-{i}$	$0$	${i}=6, 7$
$v_{{i} }$	$13-{i}$	$12-{i}$	$0$	${i}=8$
$v_{{i} }$	$13-{i}$	${i}-6$	$0$	${i}=9, 10, 11$
$v_{{i} }$	${i}-11$	${i}-6$	$0$	${i}=12, ..., 15$
$v_{{i} }$	${i}-14$	$25-{i}$	$0$	${i}=16$
$v_{{i} }$	${i}-14$	$24-{i}$	$0$	${i}=17$
$v_{{i} }$	${i}-13$	$|20-{i}|$	$0$	${i}=18, ..., 22$
$v_{{i} }$	$31-{i}$	$26-{i}$	$0$	${i}=23, 24$

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Each node of graph of the Pterocellin medication structure's given positions is unique and meets the specifications of the partition locating set. This demonstrated that the partition locating number is correct.

Hence, proved that $pd\left({G}_{ \rm{Pterocellin}}\right)\leq3.$ To verify this statement, we must demonstrate that $pd\left({G}_{ \rm{Pterocellin}}\right)\geq2,$ and then by contradiction, we will have $pd\left({G}_{ \rm{Pterocellin}}\right) = 2.$ This is not correct because this assertion is only applicable to path graphs.

Hence, proved that $pd\left({G}_{ \rm{Pterocellin}}\right) = 3.$

Daunorubicin structure's order and size are $\left|{V}\left({G}_{ \rm{Daunorubicin}}\right)\right| = 38,$ $\left|{E}\left({G}_{ \rm{Daunorubicin }}\right)\right| = 42,$ respectively. The node and bond set of the Daunorubicin drug structure are shown below. Furthermore, Figure 7 depicts the molecular graph of Daunorubicin and labeling employed in our findings.

Figure 7.  Graph of daunorubicin drug structure shown in the Figure 6.

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Theorem 2.3. Let ${G}$ be a graph of Daunorubicin drug structure. Then the partition locating number of ${G}$ is less than equal to $4.$

Proof. The graph's partition dimension or locating number of Daunorubicin drug structure is 4. To demonstrate this point, we picked a partition set of cardinality 4 and stated as $R\left({G}\right) = \{R_1, R_2, R_3, R_4\},$ where $R_1 = \{{v}_{1}\},$ $R_2 = \{{v}_{21}\},$ $R_3 = \{{v}_{36}\}$ and $R_4 = V({G}) \backslash \{{v}_{1}, {v}_{21}, {v}_{36}\}.$To demonstrate this statement true, we have included representations of each node of the Daunorubicin drug structure except ${v}_{1}, {v}_{21}$ and, ${v}_{36}$ in Table 3.

Table 3.  Representations of the nodes with respect to $R$.

$r({v}_{i}|\chi)$	$R_{1}$	$R_{2}$	$R_{3}$	$R_{4}$	Range of ${i}$
$v_{{i} }$	${i}-1$	${i}-2$	$16-{i}$	$0$	${i}=2, .., 10$
$v_{{i} }$	$19-{i}$	$20-{i}$	${i}-6$	$0$	${i}=11, ..., 18$
$v_{{i} }$	$23-{i}$	$22-{i}$	$32-{i}$	$0$	${i}=19$
$v_{{i} }$	$26-{i}$	$25-{i}$	$31-{i}$	$0$	${i}=20$
$v_{{i} }$	$31-{i}$	$31-{i}$	$30-{i}$	$0$	${i}=21$
$v_{{i} }$	$32-{i}$	$32-{i}$	$31-{i}$	$0$	${i}=22$
$v_{{i} }$	$29-{i}$	$30-{i}$	$38-{i}$	$0$	${i}=25, 26$
$v_{{i} }$	$32-{i}$	$33-{i}$	$37-{i}$	$0$	${i}=27$
$v_{{i} }$	$35-{i}$	$36-{i}$	$36-{i}$	$0$	${i}=28$
$v_{{i} }$	${i}-20$	${i}-19$	$33-{i}$	$0$	${i}=29, ..., 32$
$v_{{i} }$	${i}-22$	$47-{i}$	${i}-31$	$0$	${i}=33, ..., 35$
$v_{{i} }$	$51-{i}$	$52-{i}$	$41-{i}$	$0$	${i}=37$
$v_{{i} }$	$51-{i}$	$52-{i}$	$43-{i}$	$0$	${i}=38$

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Each node of graph of the Daunorubicin medication structure's given positions is unique and meets the specifications of the partition locating set. This demonstrated that the partition locating number was correct.

Hence, proved that $pd\left({G}_{ \rm{Daunorubicin}}\right)\leq4.$

Deguolin structure's order and size are $\left|{V}\left({G}_{ \rm{Deguolin}}\right)\right| = 38,$ $\left|{E}\left({G}_{ \rm{Deguolin }}\right)\right| = 35,$ respectively. The node and bond set of the Deguolin drug structure are shown below. Furthermore, Figure 8 depicts the molecular graph of Deguolin and labeling employed in our findings.

Figure 8.  Deguolin drug structure.

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Figure 9.  Chemical structure of Minocyline drug.

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Theorem 2.4. Let ${G}$ be a graph of Deguolin drug structure. Then the partition locating number of ${G}$ is less than equal to $4.$

Proof. The graph's partition dimension or locating number of Deguolin drug structure is 4. To demonstrate this point, we picked a partition locating set of cardinality 4 and stated as

$R\left({G}\right) = \{R_1, R_2, R_3, R_4\},$ where $R_1 = \{{v}_{2}\},$ $R_2 = \{{v}_{15}\},$ $R_3 = \{{v}_{28}\}$ and $R_4 = V({G}) \backslash \{{v}_{2}, {v}_{15}, {v}_{28}\}.$To demonstrate this statement true, we have included representations of each node of the Deguolin drug structure except ${v}_{2}, {v}_{15}$ and ${v}_{28}$ in Table 4.

Table 4.  Representations of the vertices with respect to $R$.

$r({v}_{i}|R)$	$R_{1}$	$R_{2}$	$R_{3}$	$R_4$	Range of ${i}$
$v_{{i} }$	${i}-2$	$11-{i}$	$13-{i}$	$0$	${i}=4, .., 8$
$v_{{i} }$	${i}-2$	${i}-7$	$13-{i}$	$0$	${i}=9, 10$
$v_{{i} }$	${i}-2$	$15-{i}$	$13-{i}$	$0$	${i}=11$
$v_{{i} }$	$22-{i}$	$15-{i}$	$13-{i}$	$0$	${i}=12$
$v_{{i} }$	$22-{i}$	${i}-15$	${i}-11$	$0$	${i}=16, ..., 20$
$v_{{i} }$	$24-{i}$	${i}-15$	${i}-11$	$0$	${i}=21, 22$
$v_{{i} }$	$26-{i}$	$33-{i}$	$36-{i}$	$0$	${i}=23, 24$
$v_{{i} }$	$29-{i}$	$34-{i}$	$38-{i}$	$0$	${i}=25, 26$
$v_{{i} }$	$32-{i}$	$33-{i}$	$35-{i}$	$0$	${i}=27$
$v_{{i} }$	$39-{i}$	$28-{i}$	$31-{i}$	$0$	${i}=29$
$v_{{i} }$	$35-{i}$	$34-{i}$	$38-{i}$	$0$	${i}=30$

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Each node of graph of the Deguolin medication structure's given positions is unique and meets the specifications of the partition locating set. This demonstrated that the partition locating number was correct.

Hence, proved that $pd\left({G}_{ \rm{Deguolin}}\right)\leq4.$

The node and bond set of the Minocyline drug structure are shown below. This Minocyline structure's order and size are $\left|{V}\left({G}_{ \rm{Minocyline}}\right)\right| = 35,$ $\left|{E}\left({G}_{ \rm{Minocyline }}\right)\right| = 38,$ respectively. Furthermore, Figure 10 depicts the molecular graph of Minocyline and labeling employed in our findings.

Figure 10.  Graph Minocyline drug structure shown in the Figure 9.

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Figure 11.  Chemical structure of Raloxifene drug.

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Theorem 2.5. Let ${G}$ be a graph of Minocyline drug structure. Then the partition locating number of ${G}$ is less than equal to $4.$

Proof. The graph's partition dimension of Minocyline drug structure less than equal to $4.$ To demonstrate this point, we picked a set of cardinality 4 and stated as $R\left({G}\right) = \{R_1, R_2, R_3, R_4\},$ where $R_1 = \{{v}_{20}\},$ $R_2 = \{{v}_{28}\},$ $R_3 = \{{v}_{33}\}$ and $R_4 = V({G}) \backslash \{{v}_{20}, {v}_{28}, {v}_{33}\}.$ To demonstrate this statement true, we have included representations of each node of the Minocyline drug structure except ${v}_{20}, {v}_{28}$ and ${v}_{33}$ in Table 5.

Table 5.  Representations of the vertices with respect to $R$.

$r({v}_{i}|R)$	$R_{1}$	$R_{2}$	$R_{3}$	$R_4$	Range of ${i}$
$v_{{i} }$	${i}+1$	$11-{i}$	${i}+2$	$0$	${i}=1, 2$
$v_{{i} }$	${i}+1$	$11-{i}$	${i}+1$	$0$	${i}=3, ..., 7$
$v_{{i} }$	${i}+1$	$13-{i}$	${i}+1$	$0$	${i}=8, 9$
$v_{{i} }$	${i}+1$	$13-{i}$	$19-{i}$	$0$	${i}=10$
$v_{{i} }$	$21-{i}$	${i}-9$	$19-{i}$	$0$	${i}=11, ..., 17$
$v_{{i} }$	${i}-15$	${i}-9$	${i}-27$	$0$	${i}=18, 19$
$v_{{i} }$	${i}-20$	${i}-10$	${i}-16$	$0$	${i}=21, 22$
$v_{{i} }$	${i}-19$	$33-{i}$	$29-{i}$	$0$	${i}=23, 24$
$v_{{i} }$	$31-{i}$	$33-{i}$	$31-{i}$	$0$	${i}=25$
$v_{{i} }$	$34-{i}$	$31-{i}$	$34-{i}$	$0$	${i}=26$
$v_{{i} }$	$37-1$	$33-{i}$	$37-{i}$	$0$	${i}=27$
$v_{{i} }$	${i}-18$	${i}-28$	${i}-20$	$0$	${i}=29, 30$
$v_{{i} }$	$40-{i}$	$37-{i}$	$37-{i}$	$0$	${i}=31$
$v_{{i} }$	$38-1$	$40-{i}$	$36-{i}$	$0$	${i}=32$
$v_{{i} }$	$39-{i}$	$43-{i}$	$35-{i}$	$0$	${i}=34$
$v_{{i} }$	$41-{i}$	$45-{i}$	$37-{i}$	$0$	${i}=35$

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Each node of graph of the Minocyline medication structure's given positions is unique and meets the specifications of the partition locating set. This demonstrated that the partition locating number is correct.

Hence, proved that $pd\left({G}_{ \rm{Minocyline}}\right)\leq4.$

Raloxifene structure's order and size are $\left|{V}\left({G}_{ \rm{Raloxifene}}\right)\right| = 34,$ $\left|{E}\left({G}_{ \rm{Raloxifene }}\right)\right| = 38,$ respectively.The node and bond set of the Raloxifene drug structure are shown below. Furthermore, Figure 12 depicts the molecular graph of Raloxifene and labeling employed in our findings.

Figure 12.  Graph of Raloxifene drug structure shown in the Figure 11.

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Theorem 2.6. Let ${G}$ be a graph of Raloxifene drug structure. Then the partition locating number of ${G}$ is less than equal to $4.$

Proof. The graph's partition dimension of Raloxifene drug structure is 4. To demonstrate this point, we picked a set of cardinality 4 and stated as $R\left({G}\right) = \{R_1, R_2, R_3, R_4\},$ where $R_1 = \{{v}_{20}\},$ $R_2 = \{{v}_{28}\},$ $R_3 = \{{v}_{33}\}$ and $R_4 = V({G}) \backslash \{{v}_{20}, {v}_{28}, {v}_{33}\}$.To demonstrate this statement true, we have included representations of each node of the Raloxifene drug structure except ${v}_{20}, {v}_{28}$ and ${v}_{33}$ in Table 6.

Table 6.  Representations of the vertices w.r.t. $R$.

$r({v}_{i}|R)$	$R_{1}$	$R_{2}$	$R_{3}$	$R_4$	Range of ${i}$
$v_{{i} }$	${i}-1$	${i}+5$	$14-{i}$	$0$	${i}=1, 2$
$v_{{i} }$	${i}-1$	$11-{i}$	$18-{i}$	$0$	${i}=3, 4$
$v_{{i} }$	${i}-4$	$11-{i}$	$18-{i}$	$0$	${i}=5$
$v_{{i} }$	${i}-5$	$11-{i}$	$18-{i}$	$0$	${i}=6, ..., 10$
$v_{{i} }$	${i}-5$	${i}-11$	$18-{i}$	$0$	${i}=12, 13$
$v_{{i} }$	$21-{i}$	$17-{i}$	${i}-8$	$0$	${i}=14, 15$
$v_{{i} }$	${i}-7$	${i}-13$	${i}-12$	$0$	${i}=16, 17$
$v_{{i} }$	${i}-8$	${i}-14$	${i}-15$	$0$	${i}=18, 19, 21, 22$
$v_{{i} }$	$36-{i}$	$30-{i}$	$28-{i}$	$0$	${i}=23, 24$
$v_{{i} }$	${i}-15$	${i}-21$	$|28-{i}|$	$0$	${i}=26, 27, 29, 30$
$v_{{i} }$	$45-{i}$	$39-{i}$	$34-{i}$	$0$	${i}=31, 32$
$v_{{i} }$	$49-{i}$	$43-{i}$	${i}-30$	$0$	${i}=34$

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Each node of graph of the Raloxifene medication structure's given positions is unique and meets the specifications of the partition locating set. This demonstrated that the metric locating number is correct.

Hence, proved that $pd\left({G}_{ \rm{Raloxifene}}\right)\leq4.$

3.   Conclusions

Breast cancer is one of the primary causes of cancer worldwide and the second biggest cause of cancer-related deaths in women (BC). Metastasis development is primarily linked to mortality. Therefore, it is crucial for public health that the mechanisms involved in metastasis formation are identified. Pollution and the chemical environment are among the risk factors that are being indicated as impacting the signaling pathways involved in the formation and growth of metastatic tumor cells. In this research, we consider various cancer drugs structures, computed the partition dimension, and proved that the partition dimension of Podophylb toxin and Pterocellin is $3$ and the upper bound of the partition dimension of Daunorubicin, Deguolin, Minocycline, and Raloxifene is $4$ as shown in our main findings section. This can help to understand the chemical structure of these various cancer drugs or formulas more deeply.

Acknowledgments

This work was supported by the National Science Foundation of China (11961021 and 11561019), Guangxi Natural Science Foundation (2020GXNSFAA159084), and Hechi University Research Fund for Advanced Talents (2019GCC005).

Conflict of interest

All authors declare no relationships, financial, or commercial conflicts of interest in this paper.