Identifying Critical Pathways in Coronary Heart Disease via Fuzzy Subgraph Connectivity

Abstract

Coronary heart disease (CHD) involves complex interactions among uncontrollable factors, controllable lifestyle factors, and clinical indicators, where relationships are inherently uncertain and imprecise. Traditional risk assessment models assume precise probabilistic relationships that may not capture the ambiguity in medical data. This paper introduces fuzzy subgraph connectivity (FSC) as a systematic framework for modeling uncertainty in CHD risk prediction. We construct a fuzzy graph where vertices represent risk factors and clinical indicators, with edge weights quantifying the strength of their associations. FSC measures connectivity between subgraphs to identify the strongest diagnostic pathways, dominant risk factors, and critical bridges whose removal significantly impacts predictive accuracy. Theoretical results establish bounds on connectivity and characterize conditions for maximum inter-subgraph relationships. Application to a CHD model demonstrates that FSC reveals clinically meaningful pathways, quantifies the relative influence of controllable versus uncontrollable factors, and provides interpretable insights for targeted interventions. This approach complements existing risk models by offering a transparent, graph-based framework for uncertainty quantification in cardiovascular risk assessment.

Keywords

fuzzy graph fuzzy subgraph connectivity coronary heart disease medical decision support uncertainty modeling

1. Introduction

Coronary heart disease (CHD) is a leading global health burden, accounting for significant morbidity and mortality. The identification and analysis of CHD risk factors is a major challenge in preventive cardiology. These risk factors are broadly classified into uncontrollable factors such as age and family history, and controllable factors such as smoking, diet, and physical activity. Additionally, clinical indicators such as blood pressure, cholesterol levels, and electrocardiogram (ECG) findings serve as intermediate measures linking lifestyle and hereditary factors with disease outcomes.

Conventional diagnostic methods, including regression and probabilistic models, often assume precise relationships between variables. However, in real-world medical data, relationships are rarely crisp or deterministic. For example, the effect of age on CHD varies across populations, and the influence of smoking on cardiovascular risk may depend on duration, intensity, and co-occurring conditions. These inherent uncertainties motivate the application of mathematical frameworks capable of handling imprecise, uncertain, and approximate relationships.

Fuzzy graph theory provides such a framework. The foundational concept of fuzzy sets introduced by Zadeh (Zadeh, 1965) laid the groundwork for fuzzy graph theory, formalized by Rosenfeld (Rosenfeld, 1977) . Fuzzy graphs naturally model uncertain relationships by assigning membership values in $[0, 1]$ to both vertices and edges, representing degrees of relevance and strength of connection. This approach has been successfully applied to model uncertainty in various domains, including network vulnerability (Ali et al., 2018), flow analysis (Ali et al., 2024), and health consequence modeling (Mordeson et al., 2017). Comprehensive treatments of fuzzy graph theory can be found in (Mordeson & Nair, 2000; Mathew et al., 2018), including detailed discussions on connectivity (Mathew & Sunitha, 2010). Further developments in interval-valued, vague, and domination-based fuzzy graph structures can be found in (Rashmanlou et al., 2018; Rashmanlou et al., 2016; Talebi & Rashmanlou, 2019). Related structural problems in graph theory have been explored in earlier work such as Asano and Hirata (Asano & Hirata, 1983).

In this study, we construct a fuzzy CHD graph in which vertices represent uncontrollable, controllable, and indicator factors, while edges denote their relationships with membership values in $[0, 1]$ . We introduce fuzzy subgraph connectivity (FSC) to quantify the strength of association between different categories of risk factors. Specifically, we investigate:

pairwise connectivity ( $C O N N_{G} (u, v)$ ) between individual risk factors,

vertex-to-subgraph connectivity ( $C O N N_{G} (x, H)$ ), which evaluates the influence of a single factor on a group of related components, and

subgraph-to-subgraph connectivity ( $C O N N_{G} (H_{i}, H_{j})$ ), which assesses the global strength of interaction between categories of factors.

Our analysis shows that FSC not only quantifies the strength of risk pathways but also identifies critical bridges- edges whose removal significantly reduces connectivity. Such bridges highlight key clinical factors (e.g., Holter monitoring – activity recommendations) that dominate diagnostic predictions. Moreover, strongest paths reveal the most significant diagnostic routes, providing interpretability for clinical decision-making. By bounding connectivity between the weakest and strongest observed correlations, the framework ensures reliable upper and lower limits for risk prediction. The remainder of this paper is organized as follows. Section 2 presents preliminaries on fuzzy graphs and connectivity. Section 3 develops the theory of fuzzy subgraph connectivity. Section 4 applies FSC to CHD risk prediction, including membership value justification and comparison with existing models. Section 5 concludes with future directions.

2. Preliminaries

In this section, we briefly recall the basic concepts of fuzzy sets, fuzzy graphs, and fuzzy connectivity that will be used in the sequel. Throughout, we follow standard notations used in the literature (Mordeson et al., 2018; Ali et al., 2021; Bhutani & Rosenfeld, 2003; Bhutani & Rosenfeld, 2003; Bhattacharya, 1987).

A fuzzy set A on a universe X is defined by a membership function $μ_{A} : X \to [0, 1]$ , where $μ_{A} (x)$ represents the degree of membership of x in A. For example, if X is the set of possible ages of patients, then the fuzzy set “elderly” may assign $μ_{A} (65) = 0.8$ , $μ_{A} (50) = 0.4$ , reflecting vagueness in age categorization. This concept provides a natural tool to model uncertainty in medical datasets.

A fuzzy graph G is defined as a pair $G = (σ, μ)$ , where $σ : V \to [0, 1]$ assigns a membership value to each vertex $v \in V$ and $μ : V \times V \to [0, 1]$ assigns a membership value to each edge $(u, v)$ , subject to the condition

\begin{aligned} μ (u, v) \leq min {σ (u), σ (v)} . \end{aligned}

Here $σ (u)$ may represent the relevance of a CHD factor, while $μ (u, v)$ denotes the strength of relationship between two factors. This generalization of classical graphs was first introduced by Rosenfeld (Rosenfeld, 1977). A path in a fuzzy graph G is a sequence of vertices $u_{0}, u_{1}, \dots, u_{k}$ such that $μ (u_{i}, u_{i + 1}) > 0$ for all i. The strength of a path P is defined as

\begin{aligned} str (P) = min_{(u_{i}, u_{i + 1}) \in P} μ (u_{i}, u_{i + 1}) . \end{aligned}

The u – $v$ connectivity in G is then given by

\begin{aligned} C O N N_{G} (u, v) = max_{P : u v} str (P), \end{aligned}

where the maximum is taken over all paths connecting u and v. Throughout this paper, we use the following notation:

$V$ denotes the vertex set of a fuzzy graph $G = (σ, μ)$

$σ^{*} = {v \in V : σ (v) > 0}$ denotes the support of $σ$ (vertices with non-zero membership)

$μ^{*} = {(u, v) \in V \times V : μ (u, v) > 0}$ denotes the support of $μ$ (edges with non-zero membership)

For a fuzzy subgraph $H = (ν, τ)$ , we similarly define $ν^{*}$ and $τ^{*}$

We assume all fuzzy graphs are finite, i.e., $| V | < \infty$ , ensuring that maxima and minima in our definitions exist and are attained

Some basic results used in this paper are summarized below:

Symmetry: $C O N N_{G} (H_{i}, H_{j}) = C O N N_{G} (H_{j}, H_{i})$ .

Bounds: $r (G) \leq C O N N_{G} (H_{i}, H_{j}) \leq d (G)$ , where $r (G)$ and $d (G)$ are the minimum and maximum edge strengths in G.

Bridge property: If $u v$ is a fuzzy bridge in G, then there exist subgraphs $H_{i}, H_{j}$ such that $C O N N_{G} (H_{i}, H_{j}) = μ (u v)$ .

Strongest path: The strongest path between two subgraphs corresponds to the most significant diagnostic route in an applied setting.

Definition 2.1.
Let $G = (σ, μ)$ be a fuzzy graph. A fuzzy subgraph $H = (ν, τ)$ of G is called:
Proper if $ν^{} \subset σ^{}$ (strict subset), i.e., $ν^{} \neq \emptyset$ and* $ν^{} \neq σ^{}$ ;

Induced if $τ (u, v) = μ (u, v)$ for all $u, v \in ν^{}$ ;

Disjoint* from another subgraph $H^{'} = (ν^{'}, τ^{'})$ if $ν^{} \cap ν^{' } = \emptyset$ .

Definition 2.2.
In a fuzzy graph $G = (σ, μ)$ , the distance $d (u, v)$ between vertices u and v is defined as the length (number of edges) of the shortest path connecting them in the underlying crisp graph $G^{} = (σ^{}, μ^{})$ . If no path exists, we set* $d (u, v) = \infty$ .
Definition 2.3.
Let $G = (σ, μ)$ be a finite fuzzy graph. The strength of connectedness between vertices u and $v$ , denoted by $CON N_{G} (u, v)$ , is defined as the maximum of the strengths of all paths between u and $v$ : $CON N_{G} (u, v) = max_{P : u v} str (P),$ where the path strength is $str (P) = mi n_{e \in P} μ (e)$ , and the maximum is taken over all paths P connecting u and $v$ . Since G is finite, this maximum exists and is attained.

These concepts will be applied to the fuzzy CHD graph.
3. Fuzzy Subgraph Connectivity

Definition 3.1.
$u - v$ connectivity: Maximum of the strengths of all paths between u and v is called strength of connectedness between u and v and denoted by $C O N N_{G} (u, v)$ .
Definition 3.2.
Let $G = (σ, μ)$ be a fuzzy graph with proper fuzzy subgraph $H (ν, τ)$ . For $x \in σ^{} ∖ ν^{}$ , $x - H$ connectivity is defined as maximum of strength of connectedness between x and u where $u \in ν^{}$ , denoted by* $C O N N_{G} (x, H)$ . That is $C O N N_{G} (x, H) = max_{u \in ν^{}} C O N N_{G} (x, u)$ .
Example 3.3.
Let* $G = (σ, μ)$ be a fuzzy graph with $σ^{} = {a, b, c, d, e}$ , $μ (a, b) = 0.4$ , $μ (b, c) = 0.15$ , $μ (b, d) = μ (c, d) = 0.1$ , $μ (a, d) = 0.9$ , and* $μ (e, d) = 0.3$ (see Figure 1 (a)). Let $H = (ν, τ)$ be the fuzzy subgraph induced by $ν^{} = {b, c, d}$ (see* Figure 1 (b)).

For $a \in σ^{} ∖ ν^{}$ , $\begin{array}{r} CON N_{G} (a, H) & = max {CON N_{G} (a, b), \\ CON N_{G} (a, c), CON N_{G} (a, d)} \\ = max {0.4, 0.15, 0.9} = 0.9. \end{array}$

For $e \in σ^{} ∖ ν^{}$ , vertex e connects to H only through $d$ . The paths from e to $H = {b, c, d}$ are:
$e$ – $d$ : strength $= 0.3$

$e$ – d – $b$ : strength $= min {0.3, 0.1} = 0.1$

$e$ – d – $c$ : strength $= min {0.3, 0.1} = 0.1$
Therefore, $CON N_{G} (e, H) = max {0.3, 0.1, 0.1} = 0.3.$

In the CHD context, vertex e might represent an uncontrollable factor such as age, while H represents clinical indicators. The value $CON N_{G} (e, H) = 0.3$ quantifies the diagnostic strength from this factor to the indicator panel.

Figure 1.
Fuzzy graphs G with proper induced fuzzy subgraph H.
Definition 3.4.
Let $G = (σ, μ)$ be a fuzzy graph with proper disjoint induced fuzzy subgraphs $H_{1} = (ν_{1}, τ_{1})$ and $H_{2} = (ν_{2}, τ_{2})$ . The fuzzy subgraph connectivity (FSC) between $H_{1}$ and $H_{2}$ , denoted by $CON N_{G} (H_{1}, H_{2})$ , is defined as
$\begin{aligned} CON N_{G} (H_{1}, H_{2}) = max_{u \in ν_{1}^{}, v \in ν_{2}^{}} CON N_{G} (u, v), \end{aligned}$
where $CON N_{G} (u, v)$ denotes the path-based connectivity (maximum of minimum edge strengths over all u - $v$ paths) as defined in Definition 2.3.

Throughout this paper, we consistently use the path-based definition of connectivity, where $CON N_{G} (u, v) = ma x_{P} mi n_{e \in P} μ (e)$ . This differs from an edge-based definition that would take the maximum over direct edges only.

Consider the fuzzy graph G in Figure 1 $(a)$ . Let $H_{1}$ and $H_{2}$ be fuzzy subgraphs of G induced by ${a, d}$ and ${b, c}$ respectively (see Figure 2 $(a)$ and $(b))$ . Then $C O N N_{G} (H_{1}, H_{2}) = max {C O N N_{G} (a, H_{2}), C O N N_{G} (d, H_{2})} = 0.4$ .

Figure 2.
Proper induced fuzzy subgraphs of fuzzy graph G in example 3.3.
Proposition 3.5.
Let $G = (σ, μ)$ be a fuzzy graph with proper disjoint induced fuzzy subgraphs $H_{1}$ and $H_{2}$ . Then FSC is symmetric.

For $u, v \in σ^{}$ , $C O N N_{G} (u, v) = C O N N_{G} (v, u)$ . So it is clear that $C O N N_{G} (H_{1}, H_{2})$ = $C O N N_{G} (H_{2}, H_{1})$ .

Fuzzy subgraph connectivity need not be transitive. This can be observed from the following example.
Example 3.6.
Let* $G = (σ, μ)$ be a fuzzy graph with $σ^{} = {a, b, c, d, e, f, g}$ , $σ (a, b) = 0.1$ , $σ (b, c) = 0.3$ , $σ (c, d) = 0.25$ , $σ (d, e) = 0.9 σ (c, f) = 0.11$ , $σ (a, c) = 0.9$ and* $σ (a, g) = 0.8$ (see Figure 3 ). Let $H_{1} = (ν_{1}, τ_{1})$ , $H_{2} = (ν_{2}, τ_{2})$ , and $H_{3} = (ν_{3}, τ_{3})$ be fuzzy subgraph induced by $ν_{1}^{} = {a, b}$ , $ν_{2}^{} = {d, e}$ and $ν_{3}^{} = {f, g}$ respectively. Then* ${C O N N_{G} (H_{1}, H_{2}) = 0.25$ and $C O N N_{G} (H_{2}, H_{3})} = 0.25$ . Where as $C O N N_{G} (H_{1}, H_{3})} = 0.8.$
Figure 3.
Fuzzy graph in the example 3.6.

Definition 3.7.
Let $G = (σ, μ)$ be a fuzzy graph. A pair of proper disjoint induced fuzzy subgraphs $H_{1}$ and $H_{2}$ is said to be t -fuzzy subgraph connected if $C O N N_{G} (H_{1}, H_{2}) = t$ .
Theorem 3.8.
Let $X = {H_{1}, H_{2}, \dots, H_{k}}$ be the set of fuzzy subgraphs of fuzzy graph $G = (σ, μ)$ such that $C O N N_{G} (H_{i}, H_{j}) = t$ for $i \neq j$ . Then we define a relation R on X such that $H_{i} R H_{j}$ if and only if $C O N N_{G} (H_{i}, H_{j}) = t$ or $H_{i} = H_{j}$ .

Proof. We prove that R is an equivalence relation on X by checking reflexivity, symmetry and transitivity.

For any $H_{i} \in X$ we have $H_{i} = H_{i}$ , so by definition $H_{i} R H_{i}$ . Hence R is reflexive. Let $H_{i}, H_{j} \in X$ and suppose $H_{i} R H_{j}$ . By definition this means either $H_{i} = H_{j}$ or $CON N_{G} (H_{i}, H_{j}) = t$ . If $H_{i} = H_{j}$ then trivially $H_{j} = H_{i}$ and thus $H_{j} R H_{i}$ . Otherwise, since connection is symmetric (i.e., $CON N_{G} (H_{i}, H_{j}) = CON N_{G} (H_{j}, H_{i})$ for all $i, j$ ), we have $CON N_{G} (H_{j}, H_{i}) = t$ , hence $H_{j} R H_{i}$ . Thus R is symmetric. Let $H_{i}, H_{j}, H_{k} \in X$ and assume $H_{i} R H_{j}$ and $H_{j} R H_{k}$ . We consider cases: If any two of $H_{i}, H_{j}, H_{k}$ are equal, transitivity follows immediately from reflexivity/symmetry. Otherwise all three are distinct. By hypothesis of the theorem, for every pair of distinct indices the connection equals t; in particular $CON N_{G} (H_{i}, H_{k}) = t .$ Hence $H_{i} R H_{k}$ . Therefore R is transitive.

Having checked reflexivity, symmetry and transitivity, we conclude that R is an equivalence relation on X.

Under the stated hypothesis (every pair of distinct subgraphs has connection $t$ ), every two distinct elements of X are related; hence R is actually the universal relation on X, so X is a single equivalence class under R.□
Proposition 3.9.
$u v$ is a fuzzy bridge of $G = (σ, μ)$ if and only if there exists a pair of proper induced disjoint fuzzy subgraphs $H_{1}$ and $H_{2}$ with $C O N N_{G} (H_{1}, H_{2}) = μ (u, v)$ .

Proof. Suppose $u v$ is a fuzzy bridge of G. Then removal $u v$ reduces the strength of connectedness between some pair of vertices say x and y in G. Choose $H_{1}$ as ${x}$ and $H_{2}$ as ${y}$ . Then $C O N N_{G} (H_{1}, H_{2}) = μ (u, v)$ . Conversely assume that for proper induced disjoint fuzzy subgraphs $H_{1}$ and $H_{2}$ , $C O N N_{G} (H_{1}, H_{2}) = μ (u, v)$ . Hence $u v$ is an edge of every strongest $H_{1} - H_{2}$ path in G. Choose a vertex x from $σ_{1}^{}$ and y from $σ_{1}^{}$ . It follows that $u v$ is an edge of every strongest $x - y$ path. Therefore, $u v$ is a fuzzy bridge of G.□

$P$ is a strongest path in G if and only if there exists a pair of proper disjoint induced fuzzy subgraphs $H_{1}$ and $H_{2}$ with $C O N N_{G} (H_{1}, H_{2}) = str (P)$ .
Proposition 3.10.
Let $G_{f} = (μ_{V}; μ_{E})$ be a fuzzy graph and fix a left-continuous t-norm $T$ . For a u – $v$ path $P = (u = x_{0}, x_{1}, \dots, x_{m} = v)$ define its (edge) strength by
$\begin{aligned} str (P) = T (μ_{E} (x_{0} x_{1}), μ_{E} (x_{1} x_{2}), \dots, μ_{E} (x_{m - 1} x_{m})) . \end{aligned}$

For fuzzy (induced) subgraphs $H_{1}, H_{2}$ with disjoint vertex sets, define
$\begin{aligned} CON N_{G} (H_{1}, H_{2}) = max {max_{u \in σ_{1}^{}, v \in σ_{2}^{}} max_{P \in P (u, v)} str (P)}, \end{aligned}$
i.e., the best achievable path strength between any vertex of $H_{1}$ and any vertex of $H_{2}$ . Then a path P is a strongest path in $G_{f}$ (i.e., its strength equals the maximum strength over all paths in $G_{f}$ ) if and only if there exist proper disjoint induced fuzzy subgraphs $H_{1}, H_{2}$ with $CO {NN}_{G} (H_{1}, H_{2}) = str (P)$ .

Proof. Let P be a strongest path in $G_{f}$ , and let its endpoints be u and v. Set $H_{1} = G_{f} [{u}]$ and $H_{2} = G_{f} [{v}]$ , the induced fuzzy singletons. They are proper and disjoint. By definition,
$\begin{aligned} CON N_{G} (H_{1}, H_{2}) = max_{P^{'} \in P (u, v)} str (P^{'}) = str (P), \end{aligned}$
because P is, by assumption, a strongest u – $v$ path and (being globally strongest) has strength equal to the global maximum over all paths as well. Hence the required $H_{1}, H_{2}$ exist.

Conversely, suppose there exist proper disjoint induced fuzzy subgraphs $H_{1}, H_{2}$ with $CO {NN}_{G} (H_{1}, H_{2}) = s$ . By definition of the maximum, there exist $u \in σ_{1}^{}$ , $v \in σ_{2}^{}$ , and a u – $v$ path P with $str (P) = s$ , and no path between any $u^{'} \in σ_{1}^{}$ and $v^{'} \in σ_{2}^{}$ has strength exceeding s. In particular, no path in $G_{f}$ (between any pair of vertices) can have strength $> s$ , because any such path would connect two (singletons viewed as) induced subgraphs with connection value exceeding s, contradicting maximality. Therefore P attains the global maximum of path strength in $G_{f}$ and is a strongest path.

The argument uses only that T is a monotone, associative, and (left)continuous t-norm so that (i) adding edges to a path cannot increase its T -aggregated strength and (ii) “max over paths” is well-defined and attained (or approached) by a path.□
Remark 3.11.
If we adopt the common convention $str (P) = min {μ_{E} (e) : e \in P}$ , take $T = min$ above; all steps go through verbatim. If you measure path strength with vertex memberships as well (e.g., T -aggregating both vertices and edges), replace the definition of $str (P)$ accordingly the proof structure is unchanged.
Theorem 3.12.
Let $G = (σ, μ)$ be a fuzzy graph. Then for any proper disjoint induced fuzzy subgraphs $H_{1}$ and $H_{2}$ ,
$\begin{aligned} r (G) \leq CON N_{G} (H_{1}, H_{2}) \leq d (G), \end{aligned}$
where $r (G) = mi n_{e \in μ^{}} μ (e)$ and* $d (G) = ma x_{e \in μ^{}} μ (e)$ are the minimum and maximum edge strengths in* $G$ , respectively.

Proof. Let $H_{1} = (ν_{1}, τ_{1})$ and $H_{2} = (ν_{2}, τ_{2})$ be proper disjoint induced fuzzy subgraphs of G. By Definition 7,
$\begin{aligned} CON N_{G} (H_{1}, H_{2}) = max_{u \in ν_{1}^{}, v \in ν_{2}^{}} CON N_{G} (u, v) . \end{aligned}$

For any $u \in ν_{1}^{}$ and $v \in ν_{2}^{}$ , let P be a path from u to v that achieves the maximum strength, i.e., $CON N_{G} (u, v) = str (P)$ . Then
$\begin{aligned} CON N_{G} (u, v) = str (P) = min_{e \in P} μ (e) . \end{aligned}$

Since every edge e in P satisfies $r (G) \leq μ (e) \leq d (G)$ , we have
$\begin{aligned} r (G) \leq min_{e \in P} μ (e) \leq d (G), \end{aligned}$
which gives $r (G) \leq CON N_{G} (u, v) \leq d (G)$ for all $u \in ν_{1}^{}$ and $v \in ν_{2}^{}$ .

Taking the maximum over all such pairs $(u, v)$ preserves these inequalities:
$\begin{aligned} r (G) \leq max_{u \in ν_{1}^{}, v \in ν_{2}^{}} CON N_{G} (u, v) = CON N_{G} (H_{1}, H_{2}) \leq d (G) . \end{aligned}$

This completes the proof.□

If instead one uses the path-based max–min connectivity
$\begin{aligned} conn (u, v) = max_{P : u v} min_{e \in P} μ (e), \end{aligned}$

$\begin{aligned} CON N_{G} (H_{1}, H_{2}) = max_{u \in σ_{1}^{}, v \in σ_{2}^{}} conn (u, v), \end{aligned}$
then the same inequality holds: for any path P we have $mi n_{e \in P} μ (e) \geq r (G)$ and $mi n_{e \in P} μ (e) \leq d (G)$ , hence taking the outer max over paths yields,
$\begin{aligned} r (G) \leq CON N_{G} (H_{1}, H_{2}) \leq d (G) . \end{aligned}$

So the statement is robust under both the edge-based and the usual path-based connectivity semantics.
Theorem 3.13.
Let $G = (σ, μ)$ be a fuzzy tree with proper disjoint induced fuzzy subgraphs $H_{1}$ and $H_{2}$ . Then $CON N_{G} (H_{1}, H_{2}) = κ (G)$ if and only if there exists an edge $x y \in μ^{}$ with* $x \in ν_{1}^{}$ , $y \in ν_{2}^{}$ , and $μ (x y) = κ (G)$ , where $κ (G) = ma x_{e \in μ^{}} μ (e)$ is the maximum edge strength in* G.

Proof. Let $κ (G) = ma x_{e \in μ^{}} μ (e)$ denote the maximum edge strength in G.

Assume $CON N_{G} (H_{1}, H_{2}) = κ (G)$ . By definition, there exist $u \in ν_{1}^{}$ and $v \in ν_{2}^{}$ such that $CON N_{G} (u, v) = κ (G)$ . Since G is a tree, there is a unique path P from u to v, and
$\begin{aligned} CON N_{G} (u, v) = str (P) = min_{e \in P} μ (e) . \end{aligned}$

If $CON N_{G} (u, v) = κ (G)$ , then $mi n_{e \in P} μ (e) = κ (G)$ , which means every edge in P has strength at least $κ (G)$ . Since $κ (G)$ is the maximum edge strength, all edges in P must have strength exactly $κ (G)$ . In particular, if $x y$ is the edge in P with $x \in ν_{1}^{}$ and $y \in ν_{2}^{}$ (such an edge must exist since P connects $H_{1}$ to $H_{2}$ ), then $μ (x y) = κ (G)$ .

Conversely, suppose there exists an edge $x y \in μ^{}$ with $x \in ν_{1}^{}$ , $y \in ν_{2}^{}$ , and $μ (x y) = κ (G)$ . Then
$\begin{aligned} CON N_{G} (H_{1}, H_{2}) \geq CON N_{G} (x, y) \geq μ (x y) = κ (G) . \end{aligned}$

By Theorem 3.12, $CON N_{G} (H_{1}, H_{2}) \leq κ (G)$ . Therefore, $CON N_{G} (H_{1}, H_{2}) = κ (G)$ .□
Remark 3.14.
As an immediate consequence of Theorem 3.12 and Theorem 3.13, we have $CON N_{G} (H_{1}, H_{2}) \leq κ (G)$ for any proper disjoint subgraphs $H_{1}$ and $H_{2}$ in any fuzzy graph G.

If you use the alternative “path-based” connectivity $CON N_{G} (H_{1}, H_{2}) = ma x_{P} mi n_{e \in P} μ (e)$ (where the outer max is over all paths P joining a vertex of $H_{1}$ to a vertex of $H_{2}$ ), the same inequality still holds because for any path P we have $mi n_{e \in P} μ (e) \leq κ (G)$ , and therefore $ma x_{P} mi n_{e \in P} μ (e) \leq κ (G)$ .
Theorem 3.15.
Let $G = (σ, μ)$ be a complete fuzzy graph with proper disjoint induced fuzzy subgraphs $H_{1}$ and $H_{2}$ . Then $CON N_{G} (H_{1}, H_{2}) = min {σ (u) : u \in ν_{1}^{} \cup ν_{2}^{}} .$

Proof. In a complete fuzzy graph, every pair of vertices $u, v$ is connected by an edge with $μ (u, v) = min {σ (u), σ (v)}$ .

Let $m = min {σ (u) : u \in ν_{1}^{} \cup ν_{2}^{}}$ . There exists $w \in ν_{1}^{} \cup ν_{2}^{}$ with $σ (w) = m$ . Without loss of generality, assume $w \in ν_{1}^{}$ . For any $v \in ν_{2}^{}$ ,
$\begin{aligned} μ (w v) = min {σ (w), σ (v)} = min {m, σ (v)} = m, \end{aligned}$
since $σ (v) \geq m$ by definition of m. Therefore, $CON N_{G} (w, v) \geq μ (w v) = m$ for all $v \in ν_{2}^{}$ , which gives $CON N_{G} (H_{1}, H_{2}) \geq m$ .

For any $u \in ν_{1}^{}$ and $v \in ν_{2}^{}$ , every path from u to v must pass through at least one vertex with membership m (the bottleneck vertex). Since $μ (x y) = min {σ (x), σ (y)}$ for any edge $x y$ , any path from u to v has strength at most m. Therefore, $CON N_{G} (u, v) \leq m$ for all $u \in ν_{1}^{}$ and $v \in ν_{2}^{*}$ , which gives $CON N_{G} (H_{1}, H_{2}) \leq m$ . Combining both inequalities, we obtain $CON N_{G} (H_{1}, H_{2}) = m$ .
4. Application to Chronic Heart Disease Prediction

Medical data related to Chronic Heart Disease (CHD) can be naturally modeled using fuzzy graphs, since both patient data and diagnostic indicators are inherently uncertain and imprecise. Consider a fuzzy graph representing the relationship between uncontrollable factors, indicators, and controllable factors in the context of coronary heart disease (CHD). Let the vertex set be partitioned into three types:

Uncontrollable vertices $A = {a_{1}, a_{2}, a_{3}}$ , representing factors outside direct control (e.g., age, genetics, lifestyle). Indicator vertices $C = {c_{1}, c_{2}, c_{3}, c_{4}, c_{5}, c_{6}}$ , representing measurable health indicators. Controllable vertices $D = {d_{1}, d_{2}, d_{3}, d_{4}}$ , representing factors that can be managed or intervened (e.g., blood pressure, cholesterol, exercise). Figure 4 represents a fuzzy graph model for CHD based on these data.

Figure 4.

Vertical layout of fuzzy CHD graph with uncontrollable (a_i), indicator (c_i), and controllable (d_i) data. Dotted arrows show map pings a_i → c_j, c_j →d_k, and direct a_i → d_k.

Each vertex is assigned a membership value $σ (v) \in [0, 1]$ representing its degree of contribution to CHD, while edges $u v$ are assigned fuzzy membership $μ (u v) \in [0, 1]$ representing the strength of influence or correlation between the two factors.

Thus, the entire medical model can be viewed as a fuzzy graph $G = (σ, μ)$ with $V (G) = A \cup C \cup D$ .

4.1. Fuzzy Subgraphs in the CHD Model

The fuzzy graph G naturally decomposes into the following induced fuzzy subgraphs:

$H_{1} = ⟨ A ⟩$ , $H_{2} = ⟨ C ⟩$ (indicator data), $H_{3} = ⟨ D ⟩$ .

Using our earlier definition of fuzzy subgraph connectivity (FSC):

\begin{aligned} C O N N_{G} (H_{i}, H_{j}) = max_{x \in V (H_{i})} C O N N_{G} (x, H_{j}), \end{aligned}

we can measure the relative influence between these components.

4.2. Fuzzy Membership Assignment Methodology

The fuzzy membership values $μ (a_{i} \to c_{j})$ and $μ (c_{k} \to d_{ℓ})$ in our CHD model were assigned based on the following methodology:

Clinical correlation data: We reviewed epidemiological studies, particularly the Framingham Heart Study (D’Agostino et al., 2008) and ASCVD risk guidelines (Goff Jr et al., 2014), to identify documented correlations between risk factors and clinical indicators.

In consultation with cardiologists, we normalized correlation coefficients to the $[0, 1]$ scale, where:

$μ \in [0.8, 1.0]$ : Strong, well-established relationship.

$μ \in [0.5, 0.8)$ : Moderate correlation.

$μ \in [0.3, 0.5)$ : Weak but clinically relevant association.

$μ \in [0, 0.3)$ : Minimal evidence.

$μ (a_{1} \to c_{2}) = 0.6$ : Age ( $a_{1}$ ) has a moderate correlation with stress test abnormalities ( $c_{2}$ ).

$μ (c_{4} \to d_{3}) = 0.9$ : Holter monitoring results ( $c_{4}$ ) strongly inform activity recommendations ( $d_{3}$ ).

$μ (c_{6} \to d_{4}) = 0.3$ : CT findings ( $c_{6}$ ) have weak direct influence on smoking cessation ( $d_{4}$ ) compared to other factors.

The edges represent fuzzy relationships between vertices, with membership values in

[0, 1]

indicating the strength of influence. The fuzzy edge set is defined as follows:

The resulting fuzzy CHD graph is depicted in Figure 5, showing the hierarchical structure of relationships between uncontrollable factors, indicators, and controllable factors, along with the corresponding fuzzy membership values.

Figure 5.

Fuzzy graph of CHD with vertices a_i (uncontrollable), c_i (in dicators), d_i (controllable) and fuzzy edge memberships.

The fuzzy membership values $μ (a_{i} \to c_{j})$ and $μ (c_{k} \to d_{ℓ})$ in our CHD model were assigned based on the following methodology:

In consultation with cardiologists, we normalized correlation coefficients to the $[0, 1]$ scale, where:

$μ \in [0.8, 1.0]$ : Strong, well-established relationship .

$μ \in [0.5, 0.8)$ : Moderate correlation.

$μ \in [0.3, 0.5)$ : Weak but clinically relevant association .

$μ \in [0, 0.3)$ : Minimal evidence.

$μ (a_{1} \to c_{2}) = 0.6$ : Age ( $a_{1}$ ) has a moderate correlation with stress test abnormalities ( $c_{2}$ ), supported by studies showing age-dependent cardiovascular responses (Fleisher et al., 2007).

$μ (c_{4} \to d_{3}) = 0.9$ : Holter monitoring results ( $c_{4}$ ) strongly inform activity recommendations ( $d_{3}$ ), as continuous ECG monitoring directly guides exercise prescriptions (Crawford et al., 2005).

$μ (c_{6} \to d_{4}) = 0.3$ : CT findings ( $c_{6}$ ) have weak direct influence on smoking cessation ( $d_{4}$ ) compared to other motivational factors.

Note: This is a proof concept model demonstrating the FSC methodology. Future work will validate membership values using machine learning techniques applied to large-scale patient datasets such as the UCI Heart Disease Database (Janosi & Steinbrunn, 1988).

4.3. Application of Theoretical Results

Using the fuzzy CHD graph presented above, we analyze the connectivity between the subgraphs representing uncontrollable factors ( $H_{A}$ ) and controllable factors ( $H_{D}$ ) based on the definitions of fuzzy connectivity.

$u$ - $v$ connectivity: The strength of connectedness between two vertices u and v is defined as the maximum strength of all paths connecting them, denoted by $CON N_{G} (u, v)$ . $x$ - H connectivity: For a vertex x and a fuzzy subgraph H, the x - $H$ connectivity is defined as

\begin{aligned} CON N_{G} (x, H) = max_{u \in H} CON N_{G} (x, u), \end{aligned}

representing the strongest path from x to any vertex in H.

Connectivity from $a_{2}$ to $H_{D}$ Consider the uncontrollable vertex $a_{2} \in H_{A}$ and the controllable subgraph $H_{D} = {d_{1}, d_{2}, d_{3}, d_{4}}$ . The paths from $a_{2}$ to $H_{D}$ are:

$a_{2} \to c_{3} \to d_{2}$ with edge memberships $(0.4, 0.5)$ , path strength $= min (0.4, 0.5) = 0.4$ .

$a_{2} \to d_{4}$ (direct) with membership $0.55$ , path strength $= 0.55$ .

Hence, the maximum connectivity from $a_{2}$ to $H_{D}$ is

\begin{aligned} CON N_{G} (a_{2}, H_{D}) = max {0.4, 0.55} = 0.55. \end{aligned}

The strongest path from $a_{2}$ to $H_{D}$ is the direct edge $a_{2} \to d_{4}$ , indicating that this particular controllable factor (smoking cessation, $d_{4}$ ) is most strongly influenced by the uncontrollable factor $a_{2}$ (gender). This highlights the clinical significance of $a_{2}$ in predicting and managing $d_{4}$ , which corresponds to a key controllable risk factor in CHD management. More generally, computing $CON N_{G} (x, H)$ for each uncontrollable factor x allows identification of the most influential pathways in the CHD fuzzy graph, supporting targeted interventions.

Overall connectivity between $H_{A}$ and $H_{D}$ For completeness, the overall connectivity between subgraphs $H_{A}$ and $H_{D}$ is defined as

\begin{aligned} CON N_{G} (H_{A}, H_{D}) = max_{a_{i} \in H_{A}, d_{j} \in H_{D}} CON N_{G} (a_{i}, d_{j}) . \end{aligned}

To compute this, we identify all paths from uncontrollable factors $H_{A} = {a_{1}, a_{2}, a_{3}}$ to controllable factors $H_{D} = {d_{1}, d_{2}, d_{3}, d_{4}}$ :

From $a_{1}$ : $a_{1} \to d_{3}$ : direct edge, strength $= 0.45$ . No other direct or strong two-hop paths to $H_{D}$ .

From $a_{2}$ : $a_{2} \to d_{4}$ : direct edge, strength $= 0.55$ maximum. $a_{2} \to c_{3} \to d_{2}$ : strength $= min (0.4, 0.5) = 0.4$ .

From $a_{3}$ : No direct paths to $H_{D}$ .

Therefore,

\begin{aligned} CON N_{G} (H_{A}, H_{D}) = max {0.45, 0.55, 0.4} = 0.55. \end{aligned}

The strongest path is $a_{2} \to d_{4}$ with strength $0.55$ , not the previously stated path $a_{1} \to c_{2} \to d_{1} = 0.6$ , as the edge $c_{2} \to d_{1}$ does not exist in Table 1.

Table 1.
Fuzzy Edge Membership Values.

Edge $μ_{E}$ Type

$a_{1} \to c_{2}$ 0.6 Uncontrollable $\to$ Indicator

$a_{2} \to c_{3}$ 0.4 Uncontrollable $\to$ Indicator

$a_{3} \to c_{5}$ 0.7 Uncontrollable $\to$ Indicator

$c_{1} \to d_{1}$ 0.8 Indicator $\to$ Controllable

$c_{3} \to d_{2}$ 0.5 Indicator $\to$ Controllable

$c_{4} \to d_{3}$ 0.9 Indicator $\to$ Controllable

$c_{6} \to d_{4}$ 0.3 Indicator $\to$ Controllable

$a_{1} \to d_{3}$ 0.45 uncontrollable $\to$ Controllable

$a_{2} \to d_{4}$ 0.55 Uncontrollable $\to$ Controllable

Edge	$μ_{E}$	Type
$a_{1} \to c_{2}$	0.6	Uncontrollable $\to$ Indicator
$a_{2} \to c_{3}$	0.4	Uncontrollable $\to$ Indicator
$a_{3} \to c_{5}$	0.7	Uncontrollable $\to$ Indicator
$c_{1} \to d_{1}$	0.8	Indicator $\to$ Controllable
$c_{3} \to d_{2}$	0.5	Indicator $\to$ Controllable
$c_{4} \to d_{3}$	0.9	Indicator $\to$ Controllable
$c_{6} \to d_{4}$	0.3	Indicator $\to$ Controllable
$a_{1} \to d_{3}$	0.45	uncontrollable $\to$ Controllable
$a_{2} \to d_{4}$	0.55	Uncontrollable $\to$ Controllable

This shows that among all uncontrollable factors, $a_{2}$ (gender) has the strongest overall direct influence on the controllable factors, specifically connecting to $d_{4}$ (smoking cessation). This finding suggests that gender-specific smoking cessation programs may be particularly effective in CHD risk management.

Using the fuzzy CHD graph, we verify the connectivity properties according to the previously stated propositions, theorems, and corollaries.

Symmetry (Proposition 3.5): The fuzzy subgraph connectivity (FSC) is symmetric, i.e.,

\begin{aligned} CON N_{G} (H_{1}, H_{2}) = CON N_{G} (H_{2}, H_{1}) . \end{aligned}

In our graph, the interaction between uncontrollable factors $a_{i}$ and indicators $c_{j}$ reflects this symmetry. For instance, $a_{1}$ influences $c_{2}$ with membership $0.6$ , and this bidirectional relationship indicates that indicators reflect the underlying uncontrollable factors in diagnostic prediction.

Connectivity Bounds

\begin{aligned} r (G) \leq CON N_{G} (H_{i}, H_{j}) \leq d (G), \end{aligned}

where

r (G)

and

d (G)

are the weakest and strongest fuzzy edge memberships, respectively. In our example:

\begin{aligned} r (G) = 0.3 (edge\; c_{6} \to d_{4}), d (G) = 0.9 (edge\; c_{4} \to d_{3}) . \end{aligned}

Therefore, the connectivity between any two components (e.g., controllable vs. indicator) lies in $[0.3, 0.9]$ , providing clear upper and lower limits for risk prediction.

Fuzzy bridges (Proposition 3.9): If $u v$ is a fuzzy bridge, there exist subgraphs $H_{i}, H_{j}$ such that

\begin{aligned} CON N_{G} (H_{i}, H_{j}) = μ (u v) . \end{aligned}

In practice, edges such as $a_{1} \to d_{3}$ ( $0.45$ ) or $a_{2} \to d_{4}$ ( $0.55$ ) act as critical bridges connecting uncontrollable and controllable factors directly. Removing these edges would significantly reduce the predictive connectivity in the CHD model, highlighting their importance for risk assessment.

The strongest path between subgraphs represents the most significant diagnostic route. In our graph, several clinically meaningful paths emerge:

$a_{2} \to d_{4}$ (strength $0.55$ ): Gender influences smoking cessation directly.

$a_{1} \to d_{3}$ (strength $0.45$ ): Age influences activity recommendations directly.

$a_{2} \to c_{3} \to d_{2}$ (strength $0.4$ ): Gender affects sleep patterns via echocardiogram findings

These paths provide interpretable clinical insights for targeted interventions.

Maximum edge bound:

\begin{aligned} CON N_{G} (H_{1}, H_{2}) \leq κ (G), \end{aligned}

meaning no subgraph pair can exceed the strongest individual edge. In our graph,

κ (G) = 0.9

(edge

c_{4} \to d_{3}

, Holter monitoring to activity recommendations). Hence, the most critical factor dominates the connectivity analysis, highlighting edges that should be prioritized in CHD intervention strategies.

4.4. Interpretation for CHD Prediction

The fuzzy subgraph connectivity values provide clinically meaningful interpretations:

High $CON N_{G} (H_{A}, H_{C})$ : Strong connectivity between uncontrollable factors (age, gender, family history) and medical indicators (ECG, echo, CT) implies unavoidable risk that requires close monitoring.

High $CON N_{G} (H_{C}, H_{D})$ : Strong connectivity between clinical indicators and lifestyle factors (diet, sleep, activity, smoking) suggests that preventive strategies targeting controllable factors can be effective.

Weak $CON N_{G} (H_{A}, H_{D})$ : If the direct connectivity from uncontrollable to controllable factors is weak, it reflects that lifestyle changes may not fully mitigate genetic or age-related risk, which is medically consistent with established cardiovascular risk models.

4.5. Comparison with Existing CHD Risk Models

Traditional CHD risk assessment tools include the Framingham Risk Score (D’Agostino et al., 2008) and the ASCVD Risk Calculator (Goff Jr et al., 2014), which use probabilistic regression models to predict cardiovascular events. Recent applications of fuzzy logic in cardiology (Babaoglu et al., 2010) and graph-based disease modeling (Halu et al., 2019) demonstrate growing interest in computational approaches. Our fuzzy graph approach offers complementary advantages:

Interpretability: FSC provides explicit visualization of factor interactions through graph structure and strongest paths, unlike regression coefficients that lack intuitive interpretation.

Uncertainty handling: Fuzzy memberships naturally model measurement imprecision and ambiguous clinical relationships, whereas traditional models assume precise probability distributions.

Pathway identification: FSC reveals critical diagnostic routes (strongest paths) and bottlenecks (bridges) unavailable in regression-based models, supporting mechanistic understanding.

Modular analysis: Separate assessment of uncontrollable vs. controllable factor contributions through subgraph connectivity enables targeted intervention planning.

Bridge detection: Identification of critical edges highlights key clinical factors whose modification maximally impacts risk predictions.

Hence, fuzzy subgraph connectivity provides a rigorous mathematical framework to analyze how different categories of CHD data interact. It quantifies the risk contribution of uncontrollable factors, the predictive power of medical indicators, and the preventive impact of controllable factors. By identifying strongest paths and critical bridges, clinicians can prioritize interventions that maximize risk reduction.

5. Conclusion

In this work, we developed a fuzzy graph framework for analyzing coronary heart disease risk factors. By categorizing vertices into uncontrollable (age, gender, family history), controllable (diet, sleep, activity, smoking), and indicator (ECG, stress test, echocardiogram, Holter monitoring) components with fuzzy membership values, we constructed a CHD graph modeling real-world uncertainty. Through connectivity measures—pairwise, vertex-to-subgraph, and subgraph-to-subgraph—we evaluated association strengths across components. The analysis revealed clinically meaningful results: strongest paths identify key diagnostic routes (e.g., gender to smoking cessation, $C O N N_{G} (H_{A}, H_{D}) = 0.55$ ), critical bridges highlight essential factors (Holter monitoring to activity recommendations, $μ = 0.9$ ), and connectivity bounds ( $r (G) \leq C O N N_{G} (H_{i}, H_{j}) \leq d (G)$ ) ensure reliable risk prediction limits. Compared to traditional models like Framingham Risk Score, fuzzy subgraph connectivity offers explicit pathway visualization, natural uncertainty handling, and modular factor analysis.

This study demonstrates that fuzzy subgraph connectivity captures both uncertainty and strength of medical relationships, offering interpretability for clinical decision-making. However, our current model is proof-of-concept with membership values derived from clinical guidelines. Future work will validate the framework using UCI Heart Disease Database and MIMIC-III datasets with standard metrics (sensitivity, specificity, AUC-ROC), integrate machine learning to learn membership values automatically from patient data, and extend to temporal fuzzy graphs for monitoring CHD progression over time. These developments will strengthen clinical applicability and support adoption in preventive cardiology practice.

Footnotes

Acknowledgments

The authors confirm contribution to the paper as follows: study conception, design, analysis and interpretation of results: Shanookha Ali; data collection, draft manuscript preparation: Shanookha Ali, Nitha Niralda Pandiyathumparambil Cicil. All authors reviewed the results and approved the final version of the manuscript.

ORCID iDs

Shanookha Ali

Nitha Niralda Pandiyathumparambil Cicil

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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