Abstract
Let G′ be a simple, connected, and undirected (UD) graph with the vertex set M(G′) and an edge set N(G′). In this article, we define a function as a fractional mixed dominating function (FMXDF) if it satisfies for all , where indicates the closed mixed neighbourhood of , that is the set of all such that is adjacent to and is incident with and also itself. Here, is the poundage (or weight) of f. The fractional mixed domination number (FMXDN) is denoted by and is designated as the lowest poundage among all FMXDFs of . We compute the FMXDN of some common graphs such as paths, cycles, and star graphs, the middle graph of paths and cycles, and shadow graphs. Furthermore, we compute upper bounds for the sum of the two fractional dominating parameters, resulting in the inequality , where and are the fractional edge domination number and FMXDN, respectively. Finally, we compare to other resolvability-related parameters such as metric and fault-tolerant metric dimensions on some families of graphs.
1. Introduction
Throughout this article, we use G′ as a graph with vertex set M(G′) and an edge set N(G′) and is referred to as a tree. The maximum degree is indicated by , which is the degree of the vertex with the greatest incident edges. The vertex with the lowest incidence edges is said to have the smallest degree . A vertex of degree one is known as a pendant vertex. In the case of trees, a pendant vertex is known as leaf because it has only 1 degree and it is denoted by , while its neighbour is known as a support vertex . The greatest distance between a vertex and any other node in G′ is known as the eccentricity of (see [1]). The smallest eccentricity is represented by the radius (G′) of G′. For any , the open neighbourhood of is represented as the set of all vertices adjacent to , and it is denoted by , and the closed neighbourhood of is denoted by (see [2]). Similarly, for any edge , the open and closed neighbourhoods of are defined as { such that is adjacent to } and , respectively. The set of all edges incident to as well as the vertices adjacent to is the open mixed neighbourhood of a vertex . The collection of incident vertices and as well as all the edges adjacent to is described as the open mixed neighbourhood of an edge .
Throughout this paper, we choose an element , and the open mixed neighbourhood of is indicated by , which is the set of all such that is adjacent to as well as is incident with , and the closed mixed neighbourhood of is denoted by . Recall that a set is called a dominating set of G′ if every vertex of is adjacent to at least one vertex in . The minimum cardinality of all possible dominating sets in G′ is given by the domination number (G′). Haynes et al. [3] presented the concept of multiple types of graph domination and also gave some information on fractional dominating function and mixed domination number. A set is said to be a mixed dominating (MXD) set of G′ if every element not in has at least one mixed element in . The mixed domination number (G′) of G′ is the minimum cardinality of all possible MXD sets of G′. In 1992, Sampathkumar and Kamath [4] worked on the concept of mixed domination number, in which vertices dominate edges in certain ways and vice versa.
Define a dominating function which is known as a fractional dominating function (FDF) if for every (G′). The fractional domination number (G′) is the minimum poundage (weight) among all fractional dominating functions of G′, where the poundage of is . Scheinerman and Ullman worked on the concept of fractional domination number for the minimum weight among all FDFs of G′ in [5]. Similarly, we define a dominating function which is said to be fractional edge dominating function if for every (G′). The fractional edge domination number (G′) is the minimum poundage of all fractional edge dominating functions of G′, where the poundage of is equal to (see [6]). We define a fractional mixed dominating function (FMXDF) is a function if for all M(G′) N(G′). The poundage of is given by . The fractional mixed domination number is used by the symbol of (G′), which represents the function with the least poundage among all FMXDFs of G′. For convenience, we take , , and as the fractional dominating function, fractional edge dominating function, and FMXDF of G′. For every FMXDFs, we write every closed mixed neighbourhood of an edge of G′ as for all N(G′), where star of and is the set of all incident edges of the vertex and , and it is denoted by and , and we write every closed mixed neighbourhood of a vertex of G′ as for every M(G′).
In [7–13], the authors studied the resolvability-related parameters like the metric dimension and fault-tolerant metric dimension of certain families of graphs. The classification of total dominating and fractional dominating parameters is by removing a vertex from graph G, and its related concepts are provided in [14, 15]. The concept of reinforcement number with respect to half domination number is given in [16], and Sridharan et al. [17] introduced the parameter , where , with the help of .
In recent years, various fractional dominating parameters that define the function of the domain set as either the vertex set or an edge set independently have been researched. Here, we combined the concept of a FDF and MXD set to create a function with the domain set as both a vertex and an edge set () for the minimum weight with the condition of the summation of the closed mixed neighbourhood of any element , which is at least one. In this paper, we introduce the concept of fractional mixed domination number and develop some of the results derived from this parameter. The fractional mixed domination number is greater than or equal to the fractional domination number of graph, which is the significance of this article.
In Section 2, we enumerate the FMXDN of some standard graphs, such as paths, cycles, star graphs, the middle graph of the path graphs, and the shadow graphs. In Section 3, provides the upper bound on the sum of the two parameters where fractional edge domination and FMXDN, whose resultant graph gives .
2. Some Standard Graphs
This section discusses the FMXDN of some definitive graphs such as paths, cycles, star graphs, the middle graphs of paths and cycles, and shadow graphs.
Theorem 1. If is a nontrivial path with , then .
Proof. Let G′ be a path with vertices, where represent vertices of and represent edges of . Here, , and . Let , represent elements of that are both either vertices or edges. For , obviously . For .
In Figure 1, first we assign the value 1 to and and 0 to the rest of the elements. In another way, we assign to and and (1 − ) to and and 0 to the remaining elements, where , and . For example, we assign the fractional value , and , and the remaining elements are assigned the value of zero. Next, we set the fractional value or and and the remaining elements are assigned the value of zero. Again, we set the fractional values or or and and the remaining elements are assigned the value of zero. We assign the values in this way. The only condition is when to give to and , and other and must have . For the possibilities listed above, we have . Case (i): For , where and is the positive integer. Let be any element that is either a vertex or an edge. In this case, we define a function by . For an integer . Thus, the poundage of is (G′) = . Case (ii): for , where , where as a natural number. Subcase (i): when . Similarly, we assign the value 1 to and and we give 0 to the remaining elements. This means that (G′) = is the poundage of . Another way: Alternately, to provide the function values for , we assign to and and (1 − ) to and and 0 to the remaining elements, where , and . In this way, we proceed with the paths , … to get the poundage of , where . Generally, we assign the values of the paths , … as to the elements and , and assign 1 − to the elements and , and 0 to the remaining elements, where . This implies that poundage of is (G′) = . Subcase (ii): when . We assign the value 1 to , and we give 0 to the remaining elements. Another way: Alternately, to provide the function values for , we assign , and , and the remaining elements are assigned a value of zero. In this way, we proceed with the paths , … to get the poundage of , where . In general, we assign the values of the paths , … as to the elements and , and assign 1 − to the elements and , , and 0 to the remaining elements. This implies that poundage of f is (G′) = . Case (iii): for , where . Subcase (i): In the similar process of the above cases, for , we assign 1 to and , and 0 to the rest of the elements. For , we assign 1 to , , and and 0 to the rest of the elements. Thus, the poundage of f is (G′) = . Another way: For , the values are assigned to and , and 1 − is assigned to and , and 0 is assigned to the remaining elements. For , the values are assigned to , , and and 1 − is assigned to and , and 0 is assigned to the remaining elements, where , and , and . This implies that the poundage of is (G′) = .
Theorem 2. For any cycle with , we have .
Proof. Let G′ = be a cycle graph with vertices. Let be the vertices of and be the edges of . Here, . Let x be an element of , which is either a vertex or edge. We define a function () by for all , where . In this graph, we have for every , and its poundage is the minimum among all the FMXDFs of . Thus, the poundage of is () = .
Theorem 3. For , .
Proof. Let G′ be a star graph and be vertices of G′ = , where is the central vertex of . Now, we define a function by and . This implies that the poundage of is . Suppose if we define another function by and for all ; then, obviously . As a result, is the smallest when compared to . Hence, .
Theorem 4. For any complete graph with , then , where and are size and order, respectively, and .
Proof. Let G′ = be a complete graph with . Consider that and represent the order and size of G′, and t = 2, 3, 4, … Here, we define a function by for all , where . Then, the poundage of is (G′) = . Thus, (G′) = .
Observation 5. (see [6]). For any graph , we have . Hence, it follows that and .
Corollary 6. If and for any path graph , then .
Proof. From Observation 5, we have and in this paper, for our convenience, we take the notation of FDF as . Then, . By Theorem 1, we have . Therefore, we get .
Corollary 7. For any cycle with , we have , which is always less than .
Proof. From Observation 5, we have and by Theorem 2, we have . Thus, .
Definition 8. (see [18]). The shadow graph of a connected graph G is constructed by taking two copies of G, say G′ and G″. Join each vertex of u′ of G′ to the neighbours of the corresponding vertex u″ of G″. The shadow graph of G is denoted by .
Definition 9. (see [18]). The middle graph of a connected graph G denoted by M(G) is the graph whose vertex set is where two vertices are adjacent if(i)They are adjacent edges of G(ii)One is a vertex of G and the other is an edge incident with it
Theorem 10. For , the shadow graph of is .
Proof. Consider two copies of and assume that G′ = is a shadow graph with . Let represent the vertices of the first copy of and represent the vertices of a second copy of . Let be the edges of the first copy of and be the edges of a second copy of , with connecting and and connecting and . Here, |M(G′)| = 2r and |N(G′)| = 4 (r − 1). Assume for all when r = 2. Then, the poundage of is . For , the function is defined by the following types of possibilities:(i)If , where , then we assign a value of 1 to , and f(), and a value of 0 to the remaining elements of G′.(ii)If , where , then we assign a value of 1 to , and f(), and a value of 0 to the remaining elements of G′.(iii)If , where , then we assign a value of 1 to , f(), and , and a value of 0 to the remaining elements of G′, where . Then, the poundage of is (G′) = . Hence, .
Theorem 11. The middle graph of path for is (M′ ()) = .
Proof. Let be the vertices of the path and be the edges of .
In this proof, we build the middle graph of path , that is, G′ = M′ (), with the vertex set M(G′) = and an edge set N(G′) = . Here, . In Figure 2, for M′ (), we set and 1, and all other remaining vertices are set to 0. In M′ () has , and 1, with the remaining vertices all being 0. Now, we define a function M(M′ ()) ⟶ [0, 1] bywhere . Then, the poundage of is . Hence, (M′()) = .
Theorem 12. For , (M′ ()) = .
Proof. Consider the graph G′ = M′ () with . A fractional dominating function is defined as : M(G′) ⟶ [0, 1] by () = for all , where . Thus, the poundage of is . Hence, (M′ ()) = .
Corollary 13. For any graph G′, the following holds true:(i) (G′) ≤ (G′).(ii) (G′) ≤ (G′), where and are fractional dominating and fractional edge dominating functions of G′.
Proof. Assume G′ as connected graph and as FMXDF of G′. The fractional domination number (G′) = .
Similarly, the fractional edge domination number (G′) = .
Corollary 14. For any path graph, (M′ ()) ≤ (M′ ()).
Proof. From Theorem 11 and Corollary 13, we get (M′ ()) ≤ (M′ ()).
Corollary 15. For , we have (M′ ()) < (M′ ()).
Proof. From Theorem 12 and Corollary 13, we get (M′ ()) < (M′ ()).
3. Bounds
In this section, we provide some bounds on gamma (G′) in terms of fractional domination, fractional edge domination, maximum degree , minimum degree , support s′, and leaves l′ and compare to other resolvability-related parameters.
Theorem 16. For each tree, we have and this bound is sharp.
Proof. Let be a tree with the number of vertices , the number of edges , the number of leaves , and the number of supporting vertices . For our convenience, we take the notation as and . In this theorem, is the least FMXDF of G′. A set of support vertices and leaves in a tree is represented by () and . Here, is the maximum (minimum) degree of the vertex. The vertex set can be divided into the following sets: , , , , and . Also, the edge set can be partitioned into two sets, where and . Then, the poundage of f isBy Corollary 13, we have (G′) ≤ (G′). Therefore, we have two cases here. Case (i): if , then Case (ii): if = , thenFrom (3) and (4), we get . It follows thatFor and , Theorem 16 gives the exact bound value. This completes the proof.
Theorem 17. For every tree with , then and the bound is sharp.
Proof. Let be a graph with no cycles, with the vertex as , the edge as , the maximum degree as , and the radius as . Take to be the minimal FMXDF of . The degree of an edge is described as . The vertex set can be divided into the following sets: , , , , , and . The edge set can be divided as , , , , , and . Then, the poundage of isBy Corollary 13, we have (G′) ≤ (G′). Therefore, we have two cases here. Case (i): if < , then Case (ii): if = , thenFrom Case (i) and Case (ii), we get . For and , Theorem 17 gives the exact bound value.
Definition 18. (see [10]). The length of the shortest path between a pair is said to be the distance between them. Let where and and let . The distance representation is the vector of distances from to . Such a set is called a resolving set in . A resolving set of minimum cardinality is said to be a metric dimension of , and it is denoted by . If is also a resolving set for any , then is called a fault-tolerant resolving set of . The smallest cardinality of the fault-tolerant resolving set is said to be the fault-tolerant metric dimension, and it is denoted as . For our convenience, is the metric dimension (fault-tolerant metric dimension) instead of , respectively.
Definition 19 (see [7]). The graph honeycomb rectangular torus, denoted by , is a graph constructed with where is the number of rows and is the number of columns. The order and size of this graph are and , respectively, where . The graph is 3-regular graph. The vertex and edge sets are given, respectively, as ,.
Theorem 20 (see [7]). Let be a rectangular honeycomb structure with . Then, .
Theorem 21 (see [7]). Let be a rectangular honeycomb structure with . Then, .
Theorem 22. If is a rectangular honeycomb structure with , then .
Proof. Let be a honeycomb rectangular torus with and also assume as a minimal FMXDF of . In this theorem, we put the function value for each and every element of . Therefore, the poundage of is .
Theorem 23 (see [9]). A connected graph has metric dimension 1 if and only if it is the path graph.
Theorem 24 (see [11]). A graph has if and only if it is the path graph.
Theorem 25 (see [13]). For integer , let be the -dimensional ring network. Then, 3.
Theorem 26 (see [8]). Let be the complete -partite graph with and . Then, .
Theorem 27 (see [8]). Let be the complete -partite graph with and . Then, .
Theorem 28. If is the complete -partite graph with , thenwhere and represent the cardinality of the vertex set and edge set of .
Proof. Consider as a complete -partite graph with , and also assume as a minimal FMXDF of . The cardinality of the vertex and edge sets can be represented as and , respectively. Case (i): if , then we have to find the FMXDN of with . Subcase (i): Suppose . Thus, we assign the value 1 for a maximum degree vertex and 0 to the rest of the elements of . Then, the poundage is . Subcase (ii): Suppose . Then, we assume the value 1 for all maximum degree vertices and 0 for the rest of the elements of . Then, . Subcase (iii): Take . In this case, we assign the value for all , where is the minimum degree vertex of . Then, Case (ii): if , then we have to find the FMXDN of with . Subcase (i): Let . Here, assign the value 1 for a maximum degree vertex and 0 for the rest of the elements of . Then, the poundage is . Subcase (ii): Let . Now, assign the value 1 for all maximum degree vertices and 0 for the rest of the elements of . Then, . Subcase (iii): Let . In this case, we assign the value for all , where is the minimum degree vertex of . Then, Case (iii): in general, continuing the above process for , we have to find the FMXDN of with . Subcase (i): Take . Then, assign the value 1 for a maximum degree vertex and 0 for the rest of the elements of . Then, the poundage is Subcase (ii): Take . Here, we put the value 1 for all maximum degree vertices and 0 for the rest of the elements of . Then, Subcase (iii): Take . In this case, we assign the value for all , where is the minimum degree vertex of . Then,
Theorem 29. For any tree, and this bound is sharp for , where .
Proof. Let be a tree with vertices and edges. Consider as a minimal FMXDF of , and by Definition 18, define a resolving set represents the distances from to forming the resolving set of with respect to where . Now we have to compare the value of FMXDN and the resolvability parameter like metric dimension (the resolving set with minimum cardinality). By using Theorems 1 and 23, we obtain the inequality as . Suppose is a star graph with , where . By Theorem 3, we obtain . Now we have to find the resolving set for . When , the resolving set . Next, the resolving set for is . For , the resolving set is . Continuing this way we obtain the minimum resolving set for and is 1, and for . Therefore, . In this way, we create a resolving set for any tree and obtain the inequality as . Therefore, Theorem 29 is sharp for the path , where .
Table 1 provides a comparison between the FMXDN and some resolvability-related parameters of certain graphs, such as paths, cycles, honeycomb rectangular toruses, and complete multipartite graphs.
Theorem 30. For any graph , we have and this bound is sharp.
Proof. Consider as a graph with a cardinality of vertex set as as its maximum degree, and as its minimum degree. Let be the smallest FMXDF of . Here, we have two claims. Claim 1: . For all FMXDFs, we write every closed region of an edge as for all , where and denote the star of and , and we write every closed mixed neighbourhood of a vertex as for every . Then, the poundage of is . It follows that . Then, obviously, Claim 2: . In this claim, the poundage of is, . It follows that . Here ; then obviouslyFrom (10) and (11), we get . Furthermore, this bound is sharp for , , , and when is odd. Figure 3 gives the sharpness of Theorem 30.
4. Application
The fractional mixed domination is used in hospitals. In this situation, the vertex set can be as various types of seven patients. The edge set is the connection between those patients having any one of the common symptoms (like shortness of breath, fever, fever and chills, lung infection, nausea and vomiting, abdominal pain, and diarrhea (see Figure 4)).
Doctors’ aim is to reduce the patients’ disease symptoms in a minimum number of hours. Here, the fractional mixed domination number illustrates that the minimum number of hours to reduce the disease of the patients. Here, in Figure 4, = 3.
5. Conclusion
We found the exact value of the FMXDN of some standard graphs, such as paths, cycles, the middle graph of the paths and cycles, and shadow graphs. Also, some upper bounds on the sum of the two fractional dominating parameters, whose resultant graph gives this inequality , were obtained in terms of fractional edge domination and fractional mixed domination. Finally, we provided a comparison result of and other resolvability parameters such as metric and fault-tolerant metric dimension. This new parameter will be applicable for the optimization problems in our future work.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by “Regional Innovation Strategy (RIS)” through the National Research Foundation of Korea (NRF) Funded by the Ministry of Education (MOE) (2022RIS-005). Amutha, Anbazhagan, Shanthi, and Uma would like to thank RUSA Phase 2.0 (F 24-51/2014-U), DST-FIST (SR/FIST/MS-I/2018/17), and DST-PURSE Second Phase programme (SR/PURSE Phase 2/38), Govt. of India.