The geometric constraints on Filippov algebroids

1.   Introduction

Mathematical chemistry has recently presented a wide range of approaches to deal with chemical structures that underpin existing chemical notions, as well as developing and exploring fresh mathematical models of chemical phenomena and applying mathematical concepts and techniques to chemistry. So far not a big range of scientists are working on this interdisciplinary research between mathematics and chemistry, so there is huge margin to use arithmetic properties to derive and anticipate new chemical properties. Mathematical approaches are widely used in several aspects of physical chemistry, notably in thermodynamics and compound energy. After physicists revealed in the first few years of the twentieth century that the key features of chemical compounds can be predicted using quantum theory approaches, a significant need for math in chemistry arose. The real discouragement to use math and its concepts into chemistry laboratories was the realization that chemistry cannot be understood without knowledge of quantum physics, including its complicated mathematical instruments. For further study about mathematical chemistry in terms of graph theory further articles like ^[1,2,3,4,5] can be studied. A novel topic of mathematical chemistry which is known as topological indices and its literature are found in ^[6,7,8].

Chemical graph theory is a branch of mathematical science that is used to characterise the structural properties of molecules, processes, crystals, polymers, clusters, and other materials. The vertex in chemical graph theory might be an electron, an atom, a molecule, a collection of atoms, intermediates, orbitals, and many other things. Intermolecular bonding, bonded and non-bonded connections, basic reactions, and other forces like as van der Waals forces, Keesom forces, Debye forces, and so on can all be used to illustrate the relationships between vertices of a structure.

Harary graph is an interested structure due to its complexity and beautiful topology, ^[9] find the resolvability of this structure and also discussed some other properties related to resolving vertices. Kayak paddles graph obtained by connecting two basics (path and cycle) graphs. For the resolving set of this graph we refer to see ^[10], and the necklace graph with these properties are discussed in ^[11]. The resolving set for the graph obtained from categorical product of two graphs available in ^[12], where a general circulant graphs are discussed in ^[13]. The circulant graph $C_{n}(1, 2, 3)$ are discussed in ^[14] fault-tolerant resolving sets are computed. Results related to fault-tolerant resolving set on path, cycle and some basics graph are found in ^[15]. Resolvability of different graphs in terms of fault-tolerant are available in ^[16]. Interconnection networks of fault-tolerant resolvability are discussed in ^[17] and some extremal structures are discussed in ^[18]. The general graph of convex polytopes in terms of metric, fault-tolerant and partition dimension are studied here ^[19,20,21].

The idea of resolving sets proposed by Slater ^[22] later discussed by Harary and Melter ^[23]. As described in ^[24,25], metric generators permit to get different representations for chemical compounds. Precisely, they were used in pharmaceutical research for determining patterns similar to a variety of drugs ^[38]. Metric dimension has various other applications, such as robot navigation ^[27], weighing problems ^[28], computer networks ^[29], combinatorial optimization^[30], image processing, facility location problems, sonar and coastguard loran ^[22] for further detail see ^[31,32]. Due to its variety of applications the concept of metric dimension is widely used to solve many difficult problems. For resolvability parameter of different chemical structure we refer to ^[32,33,34]. For the NP-harness of these topics we refer to see ^{[36,37,38,39]}.

Given below are basic preliminaries for the concepts studied here.

Definition 1.1. ^[40] Assume ${C}$ be an associated graph of chemical structure/network, whose vertex/node set we will denote with symbol ${N}({C})$ or simply ${N},$ while ${B}({C})$ or ${B}$ is the edge/bond set, the shortest distance between two bonds ${b}_1, {b}_2\in {N}({C})$ denoted by $S_{{b}_1, {b}_2},$ and calculated by counting the number of bonds while moving through the ${b}_1-{b}_2$ path.

Definition 1.2. ^[41] Set a subset ${L}$ from ${N}({C})$ such that each node of ${N}({C})$ have unique position with respect to ${L}.$ Then the chosen subset will be known as resolving set for a graph ${C}$ or its node set ${N}({C}).$ Mathematically, set ${L} = \{{b}_1, {b}_2, \dots, {b}_{i}\}$ in general with order or cardinality ${i}.$ For a node ${b}\in{N}({C}),$ the position is represented by ${p}({b}|{L}),$ and defined as $(S_{{b}, {b}_1}, S_{{b}, {b}_2}, \dots, S_{{b}, {b}_{i}}).$ The position vector is actually a distance of a node from each node in the chosen subset. If all the coordinate of ${p}({b}|{L})$ for each node of ${N}({C})$ is unique then ${L}$ is a candidate for the resolving set. While the least number of members of ${L},$ is the metric dimension of graph ${C}$ and it is formally shown by $dim({C}).$

Definition 1.3. Assuming that any of the member of resolving set ${L}$ is not working or any of the node from ${i}$ members, is spoiled then one can not get the unique position of entire node set. To tackle this issue the definition is known as fault-tolerant resolving set which is deal with by eliminating any of the member from ${L}$ and still obtain the unique position of entire node of a graph, symbolized as ${L}_{f}$ and the minimum members in the set denoted as $dim_{f}({C})$ and named as fault-tolerant metric dimension.

Definition 1.4. The position vector ${p}\left({b}|{L}_{p}\right) = \{(S_{b}, {L}_{p1}), \ (S_{b}, {L}_{p2}), \ \dots, \ (S_{b}, {L}_{pi})\},$ of a node $b$ with respect to ${L}_{p}.$ Where ${L}_{p}$ is the $i$-ordered proper subsets of node set and known as partition resolving set if position vector ${p}\left({b}|{L}_{p}\right)$ is unique for entire node set. The notation $pd({C})$ is the minimum count of proper subsets of $N({C})$ and called as partition dimension ^[42,43].

Theorem 1.1. ^[22] Let $dim\left({\aleph}\right),$ is the metric dimension of a graph ${\aleph}.$ Then $dim\left({\aleph}\right) = 1, \ $ iff $\ {{\aleph}}\ $ is a path graph.

2.   Construction of tripod structure ${T}\left({n}, {m}, {l}\right)$

Benzenoid systems are the natural graph representations of benzenoid hydrocarbons that is why important in theoretical chemistry. It is established fact that hydrocarbons derived from benzenoids concerning and useful in chemical industry, food industry as we in environment ^[44]. The benzenoid system we discussed here can be found in ^[45]. In which they studied, polynomial types discussion of different catacondensed and pericondensed benzenoid structures. The benzenoid tripod structure studied here is a pericondensed system. It has $4\left({n}+{m}+{l}\right)-8$ nodes and $5\left({n}+{m}+{l}\right)-11$ bonds, with all the running parameters ${n}, {m}, {l}\geq2.$ Moreover, for detailed topological study of benzenoid structures are available in ^[46]. Following are node and bonds or vertex and edge set for the benzenoid structure ${T}\left({n}, {m}, {l}\right).$ We use the labeling of nodes and edges defined in Figure 1 in our main results.

Lemma 2.1. If ${T}\left({n}, {m}, {l}\right)$ is a graph of benzenoid tripod with ${n}, {m}, {l}\geq2,$ then the minimum members in its resolving set are two.

Proof. The total number of nodes in the corresponding graph of benzenoid tripod with ${n}, {m}, {l}\geq2,$ are $4\left({n}+{m}+{l}\right)-8,$ and to check the possibilities of resolving set with cardinality two are ${C}\left(4\left({n}+{m}+{l}\right)-8, 2\right) = \frac{\left(4\left({n}+{m}+{l}\right)-8\right)!}{2\times\left(4\left({n}+{m}+{l}\right)-10\right)!}.$ Here we are checking with cardinality two, because by Theorem 1.1, the resolving set with cardinality one is reserved for path graph only. Now due to NP-hardness of choosing resolving set, we can not find the exact number of resolving sets for a graph, but from $\frac{\left(4\left({n}+{m}+{l}\right)-8\right)!}{2\times\left(4\left({n}+{m}+{l}\right)-10\right)!}$-possibilities we choose a subset ${L}$ and defined as: ${L} = \{{a}_{1}, {b}_{1}\}.$ Now to prove this claim that ${L}$ is actually one of the candidate for the resolving set of benzenoid tripod graph or ${T}\left({n}, {m}, {l}\right),$ we will follow the Definition 1.2. To fulfill the requirements of definition, we will check the unique positions or locations of each node and the methodology is defined above in the Definition 1.2.

Positions ${p}\left({a}_{i}|{L}\right)$ with respect to ${L},$ for the nodes ${a}_{i}$ with ${i} = 1, 2, \dots, 2{l},$ are given as:

Positions ${p}\left({b}_{i}|{L}\right)$ with respect to ${L},$ for the nodes ${b}_{i}$ with ${i} = 1, 2, \dots, 2{n},$ are given as:

Positions ${p}\left({c}_{i}|{L}\right)$ with respect to ${L},$ for the nodes ${c}_{i}$ with ${i} = 1, 2, \dots, 2{m}-1,$ are given as:

Positions ${p}\left({a}^{\prime}_{i}|{L}\right)$ with respect to ${L},$ for the nodes ${a}^{\prime}_{i}$ with ${i} = 1, 2, \dots, 2{l}-3,$ are given as:

Positions ${p}\left({b}^{\prime}_{i}|{L}\right)$ with respect to ${L},$ for the nodes ${b}^{\prime}_{i}$ with ${i} = 1, 2, \dots, 2{n}-3,$ are given as:

Positions ${p}\left({c}^{\prime}_{i}|{L}\right)$ with respect to ${L},$ for the nodes ${c}^{\prime}_{i}$ with ${i} = 1, 2, \dots, 2{m}-1,$ are given as:

By the given positions ${p}\left(.|{L}\right)$ of all $4\left({n}+{m}+{l}\right)-8$-nodes of ${T}\left({n}, {m}, {l}\right)$ graph of benzenoid tripod with ${n}, {m}, {l}\geq2,$ with respect to ${L},$ are unique and no two nodes have same position ${p}.$ So we can conclude that we resolve the nodes of ${T}\left({n}, {m}, {l}\right)$ with two nodes. It is implied that the minimum members in the resolving set of ${T}\left({n}, {m}, {l}\right)$ are two.

Remark 2.1. If ${T}\left({n}, {m}, {l}\right)$ is a graph of benzenoid tripod with ${n}, {m}, {l}\geq2,$ then

Proof. From the definition of metric dimension, the concept is solemnly based on the selected subset (${L}$) chooses in such a way that the entire vertex set have unique position with respect to the selected nodes or subset. In the Lemma 2.1, we already discussed the possibility of selected subset (resolving set) and according to the definition, its minimum possible cardinality. In that lemma we choose ${L} = \{{a}_{1}, {b}_{1}\}$ as a resolving set for the graph of benzenoid tripod or ${T}\left({n}, {m}, {l}\right)$ for all the possible combinatorial values of ${n}, {m}, {l}\geq2.$ We also proved in such lemma that $\left|{L}\right| = 2$ is the least possible cardinality of resolving set for the benzenoid tripod ${T}\left({n}, {m}, {l}\right).$ It is enough for the prove of what we claim in the statement that metric dimension of benzenoid tripod is two, which completes the prove.

Lemma 2.2. If ${T}\left({n}, {m}, {l}\right)$ is a graph of benzenoid tripod with ${n}, {m}, {l}\geq2,$ then the minimum members in its fault-tolerant resolving set are four.

Proof. The total number of nodes in the corresponding graph of benzenoid tripod with ${n}, {m}, {l}\geq2,$ are $4\left({n}+{m}+{l}\right)-8,$ and to check the possibilities of fault-tolerant resolving set with cardinality four are ${C}\left(4\left({n}+{m}+{l}\right)-8, 4\right) = \frac{\left(4\left({n}+{m}+{l}\right)-8\right)!}{2\times\left(4\left({n}+{m}+{l}\right)-12\right)!}.$ Here we are checking with cardinality four, later we will also check cardinality three. Now due to NP-hardness of choosing fault-tolerant resolving set, we can not find the exact number of fault-tolerant resolving sets for a graph, but from $\frac{\left(4\left({n}+{m}+{l}\right)-8\right)!}{2\times\left(4\left({n}+{m}+{l}\right)-12\right)!}$-possibilities we choose a subset ${L}_{f}$ and defined as: ${L}_{f} = \{{a}_{1}, {b}_{1}, {a}_{2{l}}, {b}_{2{n}}\}.$ Now to prove this claim that ${L}_{f}$ is actually one of the candidate for the fault-tolerant resolving set of benzenoid tripod graph or ${T}\left({n}, {m}, {l}\right),$ we will follow the Definition 1.3. To fulfill the requirements of definition, we will check the unique positions or locations of each node and the methodology is defined above in the Definition 1.3.

Positions ${p}\left({a}_{i}|{L}_{f}\right)$ with respect to ${L}_{f},$ for the nodes ${a}_{i}$ with ${i} = 1, 2, \dots, 2{l},$ are given as:

Positions ${p}\left({b}_{i}|{L}_{f}\right)$ with respect to ${L}_{f},$ for the nodes ${b}_{i}$ with ${i} = 1, 2, \dots, 2{n},$ are given as:

Positions ${p}\left({c}_{i}|{L}_{f}\right)$ with respect to ${L}_{f},$ for the nodes ${c}_{i}$ with ${i} = 1, 2, \dots, 2{m}-1,$ are given as:

Positions ${p}\left({a}^{\prime}_{i}|{L}_{f}\right)$ with respect to ${L}_{f},$ for the nodes ${a}^{\prime}_{i}$ with ${i} = 1, 2, \dots, 2{l}-3,$ are given as:

Positions ${p}\left({b}^{\prime}_{i}|{L}_{f}\right)$ with respect to ${L}_{f},$ for the nodes ${b}^{\prime}_{i}$ with ${i} = 1, 2, \dots, 2{n}-3,$ are given as:

Positions ${p}\left({c}^{\prime}_{i}|{L}_{f}\right)$ with respect to ${L}_{f},$ for the nodes ${c}^{\prime}_{i}$ with ${i} = 1, 2, \dots, 2{m}-1,$ are given as:

On the behalf of given fact for the fulfillment of definition of fault-tolerant resolving set, we can say that ${L}_{f}$ with cardinality four is possible, but when it comes to optimized value of $\left|{L}_{f}\right|,$ we still need to investigate about the minimize value of $\left|{L}_{f}\right|.$ Following are some possible cases to check that whether $\left|{L}_{f}\right| = 3$ is possible or not. Though we find the fault-tolerant resolving set with the help of algorithm and satisfied that $\left|{L}_{f}\right|\neq3,$ but for the proving purpose we build some general cases and try to conclude that only $\left|{L}_{f}\right| > 3$ is possible.

Case 1: Assume that ${L}^{\prime}_{f}\subset\{{a}_{{i}}:\ {i} = 1, 2, \dots, 2{l}\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({a}_{{r}}|{L}^{\prime}_{f}\right) = p\left({a}_{s}^{\prime}|{L}^{\prime}_{f}\right),$ where $1\leq{r}\leq2{l}$ and $1\leq{s}\leq2{l}-3.$

Case 2: Assume that ${L}^{\prime}_{f}\subset\{{b}_{{i}}:\ {i} = 1, 2, \dots, 2{n}\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({a}_{{r}}|{L}^{\prime}_{f}\right) = p\left({a}_{s}^{\prime}|{L}^{\prime}_{f}\right),$ where $1\leq{r}\leq2{l}$ and $1\leq{s}\leq2{l}-3.$

Case 3: Assume that ${L}^{\prime}_{f}\subset\{{c}_{{i}}:\ {i} = 1, 2, \dots, 2{m}-1\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({a}_{{r}}|{L}^{\prime}_{f}\right) = p\left({b}_{s}|{L}^{\prime}_{f}\right),$ where $1\leq{r}\leq2{l}$ and $1\leq{s}\leq2{n}.$

Case 4: Assume that ${L}^{\prime}_{f}\subset\{{a}_{{i}^{\prime}}:\ {i} = 1, 2, \dots, 2{l}-3\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({a}_{{r}}|{L}^{\prime}_{f}\right) = p\left({c}_{s}|{L}^{\prime}_{f}\right),$ where $1\leq{r}\leq2{l}$ and $1\leq{s}\leq2{m}-1.$

Case 5: Assume that ${L}^{\prime}_{f}\subset\{{b}_{{i}^{\prime}}:\ {i} = 1, 2, \dots, 2{n}-3\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({a}_{{r}}|{L}^{\prime}_{f}\right) = p\left({c}_{s}^{\prime}|{L}^{\prime}_{f}\right),$ where $1\leq{r}\leq2{l}$ and $1\leq{s}\leq2{m}-1.$

Case 6: Assume that ${L}^{\prime}_{f}\subset\{{c}_{{i}^{\prime}}:\ {i} = 1, 2, \dots, 2{m}-1\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({a}_{{r}}^{\prime}|{L}^{\prime}_{f}\right) = p\left({c}_{s}|{L}^{\prime}_{f}\right),$ where $1\leq{r}\leq2{l}-3$ and $1\leq{s}\leq2{m}-1.$

Case 7: Assume that ${L}^{\prime}_{f}\subset\{{a}_{i}, {b}_{j}:\ {i} = 1, 2, \dots, 2{l}, \ {j} = 1, 2, \dots, 2{n}\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({a}_{{r}}|{L}^{\prime}_{f}\right) = p\left({a}_{s}^{\prime}|{L}^{\prime}_{f}\right),$ where $1\leq{r}\leq2{l}$ and $1\leq{s}\leq2{l}-3.$

Case 8: Assume that ${L}^{\prime}_{f}\subset\{{a}_{i}, \ {c}_{j}:\ {i} = 1, 2, \dots, 2{l}, \ {j} = 1, 2, \dots, 2{m}-1\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({b}_{{r}}|{L}^{\prime}_{f}\right) = p\left({b}_{s}^{\prime}|{L}^{\prime}_{f}\right),$ where $1\leq{r}\leq2{n}$ and $1\leq{s}\leq2{n}-3.$

Case 9: Assume that ${L}^{\prime}_{f}\subset\{{a}_{i}, \ {a}_{j}^{\prime}:\ {i} = 1, 2, \dots, 2{l}, \ {j} = 1, 2, \dots, 2{l}-3\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({b}_{{r}}|{L}^{\prime}_{f}\right) = p\left({c}_{s}|{L}^{\prime}_{f}\right),$ where $1\leq{r}\leq2{n}$ and $1\leq{s}\leq2{m}-1.$

Case 10: Assume that ${L}^{\prime}_{f}\subset\{{a}_{i}, \ {b}_{j}^{\prime}:\ {i} = 1, 2, \dots, 2{l}, \ {j} = 1, 2, \dots, 2{n}-3\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({c}_{{r}}|{L}^{\prime}_{f}\right) = p\left({c}_{s}|{L}^{\prime}_{f}\right),$ where $1\leq{r}, {s}\leq2{m}-1.$

Case 11: Assume that ${L}^{\prime}_{f}\subset\{{a}_{i}, \ {c}_{j}^{\prime}:\ {i} = 1, 2, \dots, 2{l}, \ {j} = 1, 2, \dots, 2{m}-1\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({a}_{{r}}|{L}^{\prime}_{f}\right) = p\left({a}_{s}^{\prime}|{L}^{\prime}_{f}\right),$ where $1\leq{r}\leq2{l}$ and $1\leq{s}\leq2{l}-3.$

Case 12: Assume that ${L}^{\prime}_{f}\subset\{{b}_{i}, \ {c}_{j}:\ {i} = 1, 2, \dots, 2{n}, \ {j} = 1, 2, \dots, 2{m}-1\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({a}_{{r}}|{L}^{\prime}_{f}\right) = p\left({a}_{s}^{\prime}|{L}^{\prime}_{f}\right),$ or $p\left({a}_{{r}}|{L}^{\prime}_{f}\right) = p\left({b}_{k}^{\prime}|{L}^{\prime}_{f}\right),$ where $1\leq{r}\leq2{l},$ $1\leq{s}\leq2{l}-3$ and $1\leq{k}\leq2{n}-3.$

Case 13: Assume that ${L}^{\prime}_{f}\subset\{{b}_{i}, \ {a}_{j}^{\prime}:\ {i} = 1, 2, \dots, 2{n}, \ {j} = 1, 2, \dots, 2{l}-3\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({a}_{{r}}|{L}^{\prime}_{f}\right) = p\left({c}_{s}^{\prime}|{L}^{\prime}_{f}\right),$ where $1\leq{r}\leq2{l},$ $1\leq{s}\leq2{m}-1.$

Case 14: Assume that ${L}^{\prime}_{f}\subset\{{b}_{i}, \ {b}_{j}^{\prime}:\ {i} = 1, 2, \dots, 2{n}, \ {j} = 1, 2, \dots, 2{l}-3\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({c}_{{r}}|{L}^{\prime}_{f}\right) = p\left({c}_{s}^{\prime}|{L}^{\prime}_{f}\right),$ where $1\leq{r}, {s}\leq2{m}-1.$

Case 15: Assume that ${L}^{\prime}_{f}\subset\{{b}_{i}, \ {c}_{j}^{\prime}:\ {i} = 1, 2, \dots, 2{n}, \ {j} = 1, 2, \dots, 2{m}-1\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({b}_{{r}}|{L}^{\prime}_{f}\right) = p\left({b}_{s}|{L}^{\prime}_{f}\right),$ where $1\leq{r}, {s}\leq2{n}.$

Case 16: Assume that ${L}^{\prime}_{f}\subset\{{c}_{i}, \ {a}_{j}^{\prime}:\ {i} = 1, 2, \dots, 2{m}-1, \ {j} = 1, 2, \dots, 2{l}-3\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({b}_{{r}}|{L}^{\prime}_{f}\right) = p\left({b}_{s}|{L}^{\prime}_{f}\right),$ where $1\leq{r}, {s}\leq2{n}.$

Case 17: Assume that ${L}^{\prime}_{f}\subset\{{c}_{i}, \ {b}_{j}^{\prime}:\ {i} = 1, 2, \dots, 2{m}-1, \ {j} = 1, 2, \dots, 2{n}-3\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({a}_{{r}}|{L}^{\prime}_{f}\right) = p\left({a}_{s}|{L}^{\prime}_{f}\right),$ where $1\leq{r}, {s}\leq2{l}.$

Case 18: Assume that ${L}^{\prime}_{f}\subset\{{c}_{i}, \ {c}_{j}^{\prime}:\ {i}, {j} = 1, 2, \dots, 2{m}-1\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({a}_{{r}}|{L}^{\prime}_{f}\right) = p\left({a}_{s}|{L}^{\prime}_{f}\right),$ where $1\leq{r}, {s}\leq2{l}.$

Case 19: Assume that ${L}^{\prime}_{f}\subset\{{a}_{i}^{\prime}, \ {b}_{j}^{\prime}:\ {i} = 1, 2, \dots, 2{l}-3, \ {j} = 1, 2, \dots, 2{n}-3\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({a}_{{r}}|{L}^{\prime}_{f}\right) = p\left({c}_{s}|{L}^{\prime}_{f}\right),$ where $1\leq{r}\leq2{l},$ $1\leq{s}\leq2{m}-1.$

Case 20: Assume that ${L}^{\prime}_{f}\subset\{{a}_{i}^{\prime}, \ {c}_{j}^{\prime}:\ {i} = 1, 2, \dots, 2{l}-3, \ {j} = 1, 2, \dots, 2{m}-1\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({b}_{{r}}|{L}^{\prime}_{f}\right) = p\left({b}_{s}^{\prime}|{L}^{\prime}_{f}\right),$ where $1\leq{r}\leq2{n},$ $1\leq{s}\leq2{n}-3.$

Case 21: Assume that ${L}^{\prime}_{f}\subset\{{b}_{i}^{\prime}, \ {c}_{j}^{\prime}:\ {i} = 1, 2, \dots, 2{n}-3, \ {j} = 1, 2, \dots, 2{m}-1\},$ with a condition according to our requirement of theorem that $|{L}^{\prime}_{f}| = 3,$ and removal of any vertex from ${L}^{\prime}_{f}$ to fulfill the definition. The result is implied in same vertex's position and contradict our assumption with the fact that $p\left({b}_{{r}}|{L}^{\prime}_{f}\right) = p\left({b}_{s}|{L}^{\prime}_{f}\right),$ where $1\leq{r}, {s}\leq2{n}.$

By the given positions ${p}\left(.|{L}_{f}\right)$ of all $4\left({n}+{m}+{l}\right)-8$-nodes of ${T}\left({n}, {m}, {l}\right)$ graph of benzenoid tripod with ${n}, {m}, {l}\geq2,$ with respect to ${L}_{f},$ having $\left|{L}_{f}\right| = 4$ are unique and no two nodes have same position ${p}.$ It is also can be accessed that by eliminating any of arbitrary nodes from ${L}_{f}$ will not effect the definition of resolving set. We also checked that the fault-tolerant resolving set ${L}_{f}$ with $\left|{L}_{f}\right| = 3$ are resulted in with two nodes have same position ${p}.$ So we can conclude that we resolve the nodes of ${T}\left({n}, {m}, {l}\right)$ with four nodes. It is implied that the minimum members in the fault-tolerant resolving set of ${T}\left({n}, {m}, {l}\right)$ are four.

Remark 2.2. If ${T}\left({n}, {m}, {l}\right)$ is a graph of benzenoid tripod with ${n}, {m}, {l}\geq2,$ then

Proof. From the definition of fault-tolerant metric dimension (same as in parent concept), the concept is solemnly based on the selected subset (${L}_{f}$) chooses in such a way that the entire vertex set have unique position with respect to the selected nodes or subset, with addition is by removal of any arbitrary single member of ${L}_{f},$ does not effect the resolvability of vertices or position of entire nodes of graph remains unique. In the Lemma 2.2, we already discussed the possibility of selected subset (fault-tolerant resolving set) and according to the definition, its minimum possible cardinality. In that lemma we choose ${L}_{f} = \{{a}_{1}, {b}_{1}, {a}_{2{l}}, {b}_{2{n}}\}$ as a fault-tolerant resolving set for the graph of benzenoid tripod or ${T}\left({n}, {m}, {l}\right)$ for all the possible combinatorial values of ${n}, {m}, {l}\geq2.$ We also proved in such lemma that $\left|{L}_{f}\right| = 4$ is the least possible cardinality of fault-tolerant resolving set for the benzenoid tripod ${T}\left({n}, {m}, {l}\right).$ It is enough for the prove of what we claim in the statement that fault-tolerant metric dimension of benzenoid tripod is four, which completes the prove.

Lemma 2.3. If ${T}\left({n}, {m}, {l}\right)$ is a graph of benzenoid tripod with ${n}, {m}, {l}\geq2,$ then the minimum numbers of subsets of its partition resolving set are three.

Proof. The total number of nodes in the corresponding graph of benzenoid tripod with ${n}, {m}, {l}\geq2,$ are $4\left({n}+{m}+{l}\right)-8,$ and to check the possible combinations given by Bell number which is $Bell\left(4\left({n}+{m}+{l}\right)-8\right) = \sum\limits_{\alpha = 0}^{4\left({n}+{m}+{l}\right)-8}{S}\left(4\left({n}+{m}+{l}\right)-8, {\alpha}\right),$ where ${S}\left(4\left({n}+{m}+{l}\right)-8, {\alpha}\right)$ is the Stirling number of second kind ^[]. $Bell\left(4\left({n}+{m}+{l}\right)-8\right)$ is the possible number of choosing partition resolving set for ${T}\left({n}, {m}, {l}\right),$ but the best and suited one are presented here and defined as ${L}_{p} = \{{L}_{{p}_1}, \ {L}_{{p}_2}, \ {L}_{{p}_3}\},$ where ${L}_{{p}_1} = \{{a}_{1}\}, \ {L}_{{p}_2} = \{{b}_{1}\}, \ {L}_{{p}_3} = {N}\left({T}\left({n}, {m}, {l}\right)\right)\backslash\{{a}_1, \ {b}_{1}\}.$ Here we are checking with cardinality three, because by Theorem 1.1, the partition resolving set with cardinality two is reserved for path graph only. Now due to NP-hardness of choosing partition resolving set, we can not find the exact number of resolving sets for a graph, but from $Bell\left(4\left({n}+{m}+{l}\right)-8\right) = \sum\limits_{\alpha = 0}^{4\left({n}+{m}+{l}\right)-8}{S}\left(4\left({n}+{m}+{l}\right)-8, {\alpha}\right),$-possibilities we choose a subset ${L}_{p}.$ Now to prove this claim that ${L}_{p}$ is actually one of the candidate for the partition resolving set of benzenoid tripod graph or ${T}\left({n}, {m}, {l}\right),$ we will follow the Definition 1.4. To fulfill the requirements of definition, we will check the unique positions or locations of each node and the methodology is defined above in the Definition 1.4.

Positions ${p}\left({a}_{i}|{L}_{p}\right)$ with respect to ${L}_{p},$ for the nodes ${a}_{i}$ with ${i} = 1, 2, \dots, 2{l},$ are given as:

where ${z}_1 = \left\{ \begin{array}{ll} 1, & \ \ \text{if}\ {i} = 1;\\ 0, & \ \ \text{otherwise}. \end{array} \right.$

Positions ${p}\left({b}_{i}|{L}_{p}\right)$ with respect to ${L}_{p},$ for the nodes ${b}_{i}$ with ${i} = 1, 2, \dots, 2{n},$ are given as:

Positions ${p}\left({c}_{i}|{L}_{p}\right)$ with respect to ${L}_{p},$ for the nodes ${c}_{i}$ with ${i} = 1, 2, \dots, 2{m}-1,$ are given as:

Positions ${p}\left({a}^{\prime}_{i}|{L}_{p}\right)$ with respect to ${L}_{p},$ for the nodes ${a}^{\prime}_{i}$ with ${i} = 1, 2, \dots, 2{l}-3,$ are given as:

Positions ${p}\left({b}^{\prime}_{i}|{L}_{p}\right)$ with respect to ${L}_{p},$ for the nodes ${b}^{\prime}_{i}$ with ${i} = 1, 2, \dots, 2{n}-3,$ are given as:

Positions ${p}\left({c}^{\prime}_{i}|{L}_{p}\right)$ with respect to ${L}_{p},$ for the nodes ${c}^{\prime}_{i}$ with ${i} = 1, 2, \dots, 2{m}-1,$ are given as:

By the given positions ${p}\left(.|{L}_{p}\right)$ of all $4\left({n}+{m}+{l}\right)-8$-nodes of ${T}\left({n}, {m}, {l}\right)$ graph of benzenoid tripod with ${n}, {m}, {l}\geq2,$ with respect to ${L}_{p},$ are unique and no two nodes have same position ${p}.$ So we can conclude that we resolve the nodes of ${T}\left({n}, {m}, {l}\right)$ by making three subsets of nodes of graph. It is implied that the minimum members in the partition resolving set of ${T}\left({n}, {m}, {l}\right)$ are three.

Remark 2.3. If ${T}\left({n}, {m}, {l}\right)$ is a graph of benzenoid tripod with ${n}, {m}, {l}\geq2,$ then

Proof. From the definition of partition dimension (same as in parent concept), the concept is solemnly based on the selected way of doing subsets (${L}_{p}$) in such a way that the entire vertex set have unique position with respect to the selected way of making subsets of entire node set. In the Lemma 2.3, we already discussed the possibility of making the subset (partition resolving set) and according to the definition, its minimum possible cardinality. In that lemma we choose ${L}_{p} = \{{L}_{{p}_1}, \ {L}_{{p}_2}, \ {L}_{{p}_3}\},$ where ${L}_{{p}_1} = \{{a}_{1}\}, \ {L}_{{p}_2} = \{{b}_{1}\}, \ {L}_{{p}_3} = {N}\left({T}\left({n}, {m}, {l}\right)\right)\backslash\{{a}_1, \ {b}_{1}\},$ as a partition resolving set for the graph of benzenoid tripod or ${T}\left({n}, {m}, {l}\right)$ for all the possible combinatorial values of ${n}, {m}, {l}\geq2.$ We also proved in such lemma that $\left|{L}_{p}\right| = 3$ is the least possible cardinality of partition resolving set for the benzenoid tripod ${T}\left({n}, {m}, {l}\right).$ It is enough for the prove of what we claim in the statement that partition dimension of benzenoid tripod is three, which completes the prove.

3.   Conclusions

It is established fact that mathematical chemistry especially graphical chemistry, made easier way to study complex networks and chemical structures in their most easiest forms. Similarly resolvability is a parameter in which entire node or edge set and sometimes both components reshaped into specific arrangement to call or accessed them. Metric dimension is also a parameter with this property to gain each node of a structure into unique form. In this work, we consider benzenoid tripod structure to achieve its resolvability and found its minimum node-resolving set. We concluded that metric, fault-tolerant-metric and partition resolving set are with constant and exact number of members for this structure. Similar to this work, many more chemical structures can be discussed in terms of resolvability parameters. Moreover, this structure can be discussed in terms of their edge based resolvability and also edge-vertex-based resolvability. These are motivation and future direction on this topic.