However, in reality, there may be situations where the data are not precise [1] and the concepts of crisp graphs are inadequate. To address such situations, Kaufman [2] introduced a fuzzy graph in which each node and edge has a fuzzy membership value (MV) that indicates the degree of belongingness. Based on this idea, the notion of a fuzzy competition graph was proposed [3] and applied to a food web under fuzzy conditions. Later, fuzzy graph extensions were developed [4, 5]. This inspired us to study the competitions [6] on mixed graphs under fuzzy settings.

Mixed graphs have both directed and undirected edges, and can capture the mixed nature of some real-world networks, such as the current Facebook network [7]. The concept of mixed graphs [8] was first introduced in 1970 and various aspects of these graphs have been studies such as coloring [9], matrices [10,11], and isomorphism [12]. Recently, Samanta et al. defined a semidirected graph [13], which is a special type of mixed graph in which two nodes can have directed and undirected edges between them. However, all these studies were based on crisp graphs. A few additional studies have been conducted [14–17].

Das et al. [18] introduced the concept of picture fuzzy competition graphs, which model the competition among entities with uncertain and ambiguous information. They also presented some generalizations and applications of these graphs in education, ecology, business, and job markets. Deva and Felix [19] proposed a novel approach for designing a decision-making trial and evaluation laboratory method in a bipolar fuzzy environment, which can handle both positive and negative degrees of membership. Additionally, they illustrated the proposed method using a supplier selection case study. Akram and Sattar [20] investigated the notion of competition graphs under complex Pythagorean fuzzy information, which is a generalization of Pythagorean and complex intuitionistic fuzzy sets. They also explored some properties and characteristics of these graphs and reported potential applications in the field of ecology. Nithyanandham and Augustin [21] developed a novel technique for prioritizing coronavirus disease 2019 (COVID-19) vaccines based on a bipolar fuzzy p-competition graph and the additive ratio assessment method. They also compared the proposed technique with existing methods and demonstrated its validity and practicability. Narayanamoorthy et al. [22] introduced the concept of regular and totally regular bipolar fuzzy hypergraphs, which are generalizations of bipolar fuzzy graphs. They proved some mathematical properties and provided examples of these hypergraphs. Karthik et al. [23] proposed a material selection model based on spherical Dombi fuzzy graphs that could handle uncertain and ambiguous information. They applied this model to select the best material for a given application and compared it with other fuzzy models. Further details on the applications of fuzzy graphs can be found in [24–38].

Therefore, we aim to study mixed graphs in fuzzy environments and propose different types of competitions for mixed graphs under this framework.

The motivation for this study was to extend the concept of the mixed graph theory to represent different types of relationships in social networks, as well as account for the uncertainty and fuzziness that may exist in these networks. We propose fuzzy mixed graphs (FMGs), which are mixed graphs in which each vertex and link has a degree of membership between 0 and 1, to capture the ambiguities in the following and connectedness. Additionally, we investigate the concept of competition on FMGs, which can model situations in which users compete for common resources or goals and explore their properties and characteristics. We applied this concept to analyze and improve health and disaster management systems by identifying the key regions and actors most affected by the spread of diseases or disasters. We aim to contribute to the field of fuzzy graph theory and its applications in real-world problems.

The novelties of this study are as follows:

• • The introduction of FMGs, which are mixed graphs in which each vertex and link has a degree of membership between 0 and 1, captures the uncertainty in social networks.

• • The investigation of competition on FMGs, which model situations where users compete for common resources or goals, and the exploration of its properties and characteristics.

• • Competition is applied to FMGs to analyze and improve health and disaster management systems. More specifically, the application of competition enables FMGs to identify the key regions and actors that are most affected by the spread of diseases or disasters.

These novelties contribute to the field of fuzzy graph theory and its applications to real-world problems.

This paper is organized as follows. Section 1 introduces the study with a literature review. Section 2 describes some basic notions related to the study. Section 3 introduces competition fuzzy mixed graph. Section 4 introduces m-step competition fuzzy mixed graph. Section 5 proposes some related properties. Section 6 describes a numerical application. Section 7 concludes the study with future directions.

2.1 Graph

Let a graph G = (V, E) having n nodes and m links are represented by an adjacent matrix A = {a_ij} ∈ R_n×n, where a_ij = 1 if node i is linked with node j and a_ij = 0 otherwise. In social networks, two nodes are related by a link.

2.2 Mixed Graph

Definition 2. Let N be a set of vertices or nodes of a graph and L = L₁ ∪ L₂ where L₁ = {(u, v) | u, v ∈ N} ⊂ N × N called a set of undirected edges of the graph and called a set of directed edges of the graph. The graph X = (N, L₁, L→₂) is then said to form a mixed graph.

2.3 Fuzzy Graph

Let G = (V, E) be a graph with the pair of mappings σ : V → [0, 1] and μ : E → [0, 1] such that the condition μ(x, y) ≤ σ(x) ∧ σ(y) for all (x, y) ∈ E ⊆ V × V, G = (V, E, μ, σ) is then called a fuzzy graph.

2.4 UndirectedWalk and Path

Definition 4. Let X = (N, L₁, L→₂, μ₁, μ₂, σ, δ) be an FMG. An undirected walk is a sequence : v₁e₁v₂e₂v₃ ... e_kv_k+1, where v₁, v₂, v₃, ..., v_k+1 are the vertices and e₁, e₂, ..., e_k are the edges, such that all the edges are undirected and μ₁(e_i) > 0 for i = 1, 2, ..., k. An undirected walk from vertex u to v is said to be an undirected path of length m if exactly m links occur in the walk and no vertices and links repeat except u, v, and it is denoted by P(u,v)m.

Example 2. Let us consider an FMG (Figure 1). From Figure 1, v₁-v₄-v₃ is an undirected path.

2.5 Directed Walk and Path

Definition 5. Let X = (N, L₁, L→₂, μ₁, μ₂, σ, δ) be an FMG. A directed walk is a sequence : v₁ e→1 v₂ e→2 v₃ ... e→k v_k+1, where v₁, v₂, v₃, ..., v_k+1 are the vertices and e→1, e→2, ..., e→k are the edges, such that all the edges are directed in the same direction, and μ₂(e→i) > 0 for i = 1, 2, ..., k. A directed walk from vertex u to v is said to be a directed path of length m if exactly m links occur in the walk and no vertices and links repeat except u, v, and it is denoted by P→(u,v)m.

Example 3. Consider an FMG (Figure 1). From Figure 1, it follows that v₃ → v₂ → v₄ is the directed path of the graph.

2.6 Neighborhood

Definition 6. Let X = (N, L₁, L→₂, μ₁, μ₂, σ, δ) be an FMG. The neighborhood of vertex u is denoted by NB(u) and is defined as the set

Note.

• • The out-neighborhood of vertex u is denoted by NB⁺(u) and is defined as the set NB+(u)={(v,μ2(u,v→)+μ2(u,v→)δ(u,v→)):v∈N and μ2(u,v→)>0}.

• • The in-neighborhood of vertex u is denoted by NB⁻(u) and is defined as the set NB-(u)={(v,μ2(u,v→)+μ2(u,v→)δ(u,v→)):v∈N and μ2(u,v→)>0}.

Example 4. Let us consider an FMG (Figure 1). Here, from Figure 1, it follows that

• (a) The neighborhood of v₁ is NB(v₁) = {(v₄, 0.4)}.

• (b) The out-neighborhood of v₁ is NB⁺(v₁) = {(v₂, 0.44)}.

• (c) The in-neighborhood of v₂ is NB⁻ (v₂) = {(v₁, 0.44), (v₃, 0.66)}.

2.7 m-Step Neighborhood

Definition 7. Let X=(N,L1,L2→,μ1,μ2,σ,δ) be an FMG. The m–step neighborhood of vertex u is denoted by NB_m(u) and is defined as the set NBm(u)={(v,σm(v)):v∈N and σm(v)=min{μ1(x,y):(x,y)∈P(u,v)m}}.

Note.

• • The m-step out-neighborhood of vertex u is denoted by NB_m⁺(u), and is defined as the set

  NBm+(u)={(v,σm(u,v)):v∈N and σm(u,v)}=min{μ2(a,b→)+μ2(a,b→)δ(a,b→):(a,b→)∈P→(u,v)m}}.

• • The m-step in the neighborhood of vertex u is denoted by NBm-(u) and is defined as the set

  NBm-(u)={(v,σm(u,v)):v∈N and σm(u,v)}=min{μ2(a,b→)+μ2(a,b→)δ(a,b→):(a,b→)∈P→(v,u)m}}.

Example 5. Let us consider an FMG (Figure 1). From Figure 1, it follows that

• (a) 2-step neighborhood of v₁ is NB₂(v₁) = {(v₃, 0.3)}.

• (b) 2-step out-neighborhood of v₃ is NB2+(v3)={(v4,0.48)}.

• (c) 2-step in-neighborhood of v₄ is NB2-(v4)={(v1,0.44),(v1,0.48)}.

3.1 Definition of CFG

Definition 8. Let X=(N,L1,L2→,μ1,μ2,σ,δ) be an FMG. The competition fuzzy graph (CFG) of X is denoted by C(X) and is defined as an undirected fuzzy graph in which N is also a vertex set. There exists an edge between vertices x, y ∈ N if NB⁺(x) ∩ NB⁺(y) ≠ ∅ with membership value α(x, y) is defined by

3.2 Algorithm to Find the CFG of FMG and the Adjacency Matrix of CFG

Algorithm 1.

• Step 1 : Consider an FMG with given memberships of vertices and edges.

• Step 2: Find the neighbors for all x_i ∈ N.

• Step 3: Find the intersections NB⁺(x_i) ∩ NB⁺(x_j) for all x_i, x_j ∈ V and x_i ≠ x_j .

• Step 4: If NB⁺(x_i) ∩ NB⁺(x_j) = {x_k : k = 1, 2, ..., }, then find μ2(xi,xm→)δ(xi,xm→)-μ2(xj,xm→)δ(xj,xm→) and same for all non-void intersections.

• Step 5: Find α(xi,xj)=minxk∈N+(xi)∩N+(xj)|μ2(xi,xm→)δ(xi,xm→)-μ2(xj,xm→)δ(xj,xm→)|. The values obtained in this section are the membership values of the edges of the CFG.

• Step 6: If a_ij denotes the ij–th element of adjacency matrix of the CFG, then

A flowchart of the algorithm is shown in Figure 2.

Example 6. Let us consider an FMG (Figure 1). Here, from Figure 1, the CFG is defined as Figure 3.

4.1 Definition of m-Step CFG

Definition 9. Let G=(N,L1,L2→,μ1,μ2,σ,δ) be an FMG. The m-step CFG of G is denoted by C_m(G) and is defined as an undirected fuzzy graph in which N is also a vertex set, and there exists an edge between vertices x, y ∈ N if NBm+(x)∩NB+(y)≠∅ with membership value β^m(x, y) defined by

4.2 Algorithm to Find the m-Step CFG of FMG and the Adjacency Matrix of m-Step CFG

Algorithm 2.

• Step 1: Consider an FMG with given memberships of vertices and edges.

• Step 2: Find the neighbors for all x_i ∈ N.

• Step 3: Find the intersections NBm+(xi)∩NBm+(xj) for all x_i, x_j ∈ N and x_i ≠ x_j.

• Step 4: If NB⁺(x_i) ∩ NB⁺(x_j) = {x_k : k = 1, 2, ..., }, then find σm(xi,xk→)-σm(xj,xk→) and same for all non-void intersections.

• Step 5: Find βm(xi,xj)=minxk∈NB+(xi)∩NB+(xj)∣σm(xi,xk→)-σm(xj,xk→)∣. The values obtained in this section are the membership values of the edges of the CFG.

• Step 6: If a_ij denotes the ij-th element of the adjacency matrix of the CFG, then

A flowchart of the algorithm is shown in Figure 4.

Example 7. Let us consider an FMG (Figure 1). From Figure 1, the 2-step CFG is defined as Figure 5.

Theorem 1. Let X = (N, L) be a fuzzy graph. Subsequently, FMG X′ with C(X′) = X. occurs.

Proof.

Given:

Here, X = (N, L) is a fuzzy graph.

To prove : C(X′) = X

Let (x, y) be an edge of a fuzzy graph. FMG X→′ is created, where C(X′) = X.

Let x′, y′ ∈ X′ be the equivalent vertices of x, y ∈ X.

We can join two directed edges x′, y′ to a vertex z′ ∈ X′ such that z′ ∈ NB⁺(x′) ∩ NB⁺(y′).

Therefore, this can be done for all nodes and links of X.

Hence, C(X′) = X.

Theorem 2. The competition number (CN) of a circuit in a fuzzy graph is 1.

Proof.

Claim:

Let us consider a circuit of length n in a fuzzy graph.

To prove:CN of a circuit in a fuzzy graph is 1.

We drew all possible directions with the vertices of the circuit to fit a corresponding FMG whose competition graph is the circuit. However, it is impossible to draw without isolating at least one extra vertex from the circuit.

Hence, CN of a circuit in a fuzzy graph is 1.

Theorem 3. If X_m is the m–step graph of graph X, then C(X_m) = C_m(X).

Proof:

Given: Here, X is an FMG and X_m is an m-step graph of graph X.

To prove : C(X_m) = C_m(X)

Here, the vertex set of X and X_m are equal.

Let us take (u, v) ∈ C(X_m).

Here, (u, v) ∈ C(X_m),; therefore, there exist edges (u,w1→),(v,w1→);(u,w2→),(u,w2→);(u,wn→),(v,wn→) of some integer n.

Here, NB⁺(u) ∩ NB⁺(v) = {w₁, w₂, ..., w_n}.

An edge (u,w1→)∈Xm indicates that there occurs a u,w1→m-length path from u to w₁ and same as for (v,w1→)∈Xm.

Hence,

Similarly, edge (x, y) ∈ C_m(X), implies that

Here, from (1) and (2), it follows that C(X_m) = C_m(X).

Note.

• • In an FMG, if a vertex has only in-neighbors, then the vertex must be isolated from the corresponding competition graph.

• • The competition graph on every fuzzy mixed circuit graph is a null graph.

Similar to competition in an ecosystem, economic competition among countries is considered here for representation. The competition between countries with respect to the health index (HI) and disaster risk index (DI) was evaluated. Therefore, we need to construct a network before evaluation and presentation.

• • The advantages of this algorithm in health and disaster management are

  • ○ It captures the uncertainty and variability in the relationships among users, regions, and resources in a realistic manner using degrees of membership instead of binary values.

  • ○ It measures the intensity of competition among users, regions, and resources using the concept of maximal competitors and the competition index.

  • ○ It identifies the most vulnerable and influential regions and actors in the network using the concepts of competition centrality and degree of competition.

  • ○ It suggests optimal strategies for reducing the spread of diseases or disasters using the concepts of competition reduction and minimization.

6.1 Construction of a Network

To construct a mixed network, we consider 10 countries : Germany, India, the United Kingdom, France, Italy, Brazil, Canada, Russia, South Korea, and Spain.

All countries compete for health and disaster management (Figure 6). Therefore, there are direct links between health and disaster. Health and disaster index data were obtained from Wikipedia. All data with normalized values are listed in Table 1.

6.2 Algorithm

The results of competitions are calculated using the following steps.

• Step 1: Consider a network of countries with health and disaster risks. All degrees of membership of the edge were considered as input.

• Step 2: Find the out-neighbors of the countries and the intersections of all pairwise out-neighbors of the countries.

• Step 3: Represent the network as FMG G = (V, E₁, E₂, μ₁, μ₂, σ), where V is the set of vertices (countries), E₁ is the set of undirected edges (health relations), E₂ is the set of directed edges (disaster relations), μ₁ and μ₂ are the membership functions for E₁ and E₂, respectively, and σ is the membership function for V. Adjacency matrices can be used to represent FMG1.

• Step 4: Find the degree of membership of competing countries toward health and disaster using Table 1.

• Step 5: The required result for the competition is obtained using Algorithm 2.

• Step 6: Calculate the fuzzy mixed degree of each vertex, which is the sum of the degrees of membership of all edges incident to the vertex. The fuzzy mixed degree reflects the overall competitive performance of a country.

• Step 7: Compare the fuzzy mixed degrees of the competing countries and rank them according to their values. The higher the fuzzy mixed degree, the better is the country’s performance.

6.3 Result

Table 2 presents the competition in health management, and Table 3 presents the competition in disaster management. The membership values for the corresponding competition graphs are listed in Table 4.

Thus, a real-life competition for health and disaster management among countries was presented using the concepts of the proposed algorithms. The following results were observed:

• 1. The lower non-zero values in Table 4 indicate that competition among the corresponding countries is higher.

• 2. The competition value should be used as the comparison between two items.

• 3. Competition studies should be used as models for data analysis in marketing.

We presented the concept of FMGs as an extension of mixed graphs, and defined and analyzed the competition and competition numbers on FMGs. We also demonstrated the application of competition on FMGs to health and disaster scenarios. This study opens new avenues for future research on various topics in FMG theory, such as interval-valued FMGs, generalized FMGs, and fuzzy mixed planar graphs, and their potential applications in science and engineering problems.

Table 1. Collections of data on health and disasters of countries from Wikipedia.

Table 2. Competition for health.

Table 3. Competition for disaster.

Table 4. Resultant competition.