An Algorithmic Approach to Bridge Detection in Bijective Neutrosophic Graph with Application in Cancer Diagnosis

Neutrosophic Sets and Systems, Vol. 96, 2026 University of New Mexico __________________________________________________________________________________________________ D.Rajalaxmi, V.Vijaya and S.Revathi, An Algorithmic Approach to Bridge Detection in Bijective Neutrosophic Graph with Application in Cancer Diagnosis An Algorithmic Approach to Bridge Detection in Bijective Neutrosophic Graph with Application in Cancer Diagnosis D.Rajalaxmi1, V.Vijaya2and S.Revathi3 1,2PG and Research Department of Mathematics, Seethalakshmi Ramaswami College, Tiruchirappalli,(Affiliated to Bharathidasan University, Tiruchirappalli), [email protected], [email protected] 3Department of Mathematics, Saranathan College of Engineering, Tiruchirappalli, India [email protected] *Correspondence: [email protected] Abstract: This paper introduces an efficient algorithm for detecting bridges in Bijective neutrosophic graphs, an emerging approach in handling uncertainty in complex systems such as medical diagnostics. The algorithm identifies T-bridges, Ibridges and F-bridges in neutrosophic graphs, and incorporates a de neutrosophication method using score function to find neutrosophic bridges when needed. The application of the algorithm to cancer diagnosis is explored, demonstrating its potential to enhance disease identification and treatment planning based on symptoms. The result suggest that this method can improve the accuracy of medical decisionmaking in uncertain environments, thereby offering a promising tool for healthcare analysis. By providing a more precise understanding of medical data, this approach has the potential to optimize diagnostic processes and treatment strategies in uncertain and indeterminate contexts. Keywords: Neutrosophic Graph; Bijective Neutrosophic graph; T-bridge, I-bridge, Fbridge; and Neutrosophic Bridge. 1. Introduction Graph theory has emerged as a powerful mathematical tool for modeling complex systems in various domains, including biology, computer science, and medicine. Among its numerous applications, the analysis of graph structures offers valuable insights into the connectivity and critical components of networks. In this context, bridge detection plays a significant role in identifying the most sensitive or pivotal connections whose removal may Neutrosophic Sets and Systems, Vol. 96, 2026 285 __________________________________________________________________________________________________ D.Rajalaxmi, V.Vijaya and S.Revathi, An Algorithmic Approach to Bridge Detection in Bijective Neutrosophic Graph with Application in Cancer Diagnosis fragment the graph. To address the limitations of classical graph models in handling uncertainty, incompleteness and inconsistency inherent in real world dataparticularly in medical diagnosticsthe concept of neutrosophic graphs has gained traction. These graphs extend fuzzy and intuitionistic fuzzy models by incorporating degrees of truth, indeterminacy, and falsity. A further advancement, bijective neutrosophic graphs, ensures a one to one correspondence between graph elements, offering a more precise and structured representation of data. Akram, M et. al [1] introduced the concept of operations on Single Valued Neutrosophic Graphs (SVN-graphs) in 2017 Beaula Thangaraj et al. [2–4] contributed significantly to the field by applying various fuzzy numbers and ranking methods to solve critical path problems, exemplifying the expanding application of fuzzy logic in optimization and decision-making. It is important to acknowledge Broumi, S et al[5-6] studies about single valued neutrosophic graph in 2016. Further this concept take its shape as neutrosophic labelling graph in 2019, which was introduced by Gomathi et.al [7]. A.Hassan et.al [8] studied about single valued trees in 2018. Muthuraj et.al[9] studied about multi fuzzy graph in 2020. Rajalaxmi D et.al[10-11] studied about metric in fuzzy labeling graph and Bijective single valued in highlighting their structural properties and potential applications. Vijaya et al[12,13] have shaped the advancements of Neutrosophic graphs in find the solution of Decision making problem and critical path problems by using Pythagorean Fuzzy numbers and Neutrosophic Fuzzy numbers. Ye, J.,[14-15]studied SingleValued Neutrosophic Minimum Spanning Tree in 2014. Finally, the pioneering work of Zadeh, L. [16] in 1965, who introduced the concept of fuzzy sets, laid the groundwork for the entire field of fuzzy and neutrosophic mathematics that followed. An algorithmic method for detecting bridges in bijective neutrosophic graphs is presented in this article, with a focus on its use in the diagnosis of cancer. In addition to improving structural analysis of intricate networks, the algorithm offers a useful tool that can be applied to a range of real-world issues. The suggested techniques seek to assist oncology's early detection, risk assessment, and strategic intervention planning by locating important pathways or interactions in biological networks linked to cancer. Through this study, we show how algorithmic approaches and mathematical abstraction can greatly advance medical research and decision-making. The neutrosophic framework permits the simultaneous representation of truth, indeterminacy, and falsity, the neutrosophic framework is especially useful in this situation. Incomplete, ambiguous, or contradicting information is frequently present in biological and medical networks, particularly those pertaining to the diagnosis of cancer. The ability of fuzzy graph models and even classical graph theory to handle such uncertainties is constrained. The suggested approach more successfully captures several aspects of uncertainty by using bijective neutrosophic graphs, producing robust analysis and more trustworthy results. This makes the neutrosophic approach especially important in advancing medical research and decision-making processes where ambiguity is inherent. 2. Preliminaries: Neutrosophic Sets and Systems, Vol. 96, 2026 286 Definition 2.1[7]: A neutrosophic graph is of the form G = (V, σ,μ ) where σ = (T1, I1, F1) and μ = (T2, I2, F2) (i) V = {v1, v2, v3, ···, vn} such that T1: V → [0, 1], I1: V → [0, 1] and F1 : V → [0, 1] denote the degree of truth-membership function, indeterminacy-membership function and falsity-membership function of the vertex vi ∈ V respectively, and 0 ≤ T1 (v) + I1 (v) + F1 (v) ≤ 3 ∀ vi ∈ V (i=1, 2, 3….n). (ii) T2 : V × V → [0, 1], I2 : V × V → [0, 1] and F2 : V × V → [0, 1], where T2(vi, vj) , I2(vi, vj) and F2(v i, vj) denote the degree of truth-membership function, indeterminacy membership function and falsity-membership function of the edge (v i, vj) respectively such that for every (vi, vj), T2 (vi, vj) ≤ min {T1(vi), T1(vj)}, I2 (vi, vj) ≤ min {I1(vi), I1(vj )}, F2 (vi, vj) ≤ max {F1 (vi), F1(vj)}, and 0 ≤ T2(vi, vj) + I2(vi, vj) + F2(vi, vj) ≤ 3 . Definition 2.2 A neutrosophic graph is said to be a bijective neutrosophic graph if ]1,0[:],1,0[:],1,0[:],1,0[:],1,0[:],1,0[: →→→→→→ VVVVVVVVV FITFIT  are bijective, such that the truthmembership function, Indeterminacy-membership function and Falsitymembership functions for every edge ) , (vu T  < min ( T  (u), T  (v)) ), (v u I  < min ( I  (u), I  (v)) ),( vu F  < max ( F  (u), F  (v)) and 0  ),( vu T  + ) ,( v u I  + ),( vu F   3 Figure 1:Bijective Neutrosophic Graph Definition :2.3 The strength of the path with n edges is defined as S(P) = (S(P1), S(P2), S(P3)) where __________________________________________________________________________________________________ D.Rajalaxmi, V.Vijaya and S.Revathi, An Algorithmic Approach to Bridge Detection in Bijective Neutrosophic Graph with Application in Cancer Diagnosis Neutrosophic Sets and Systems, Vol. 96, 2026 287 S(P1) = ), ( 1vu T n i  =  , S(P2) = ),( 1v u I n i  =  , S(P3) = ),( 1 vu n F i V  − Definition :2.4 Let G be a Bijective neutrosophic graph. The connected between any two vertices is defined by )),(),,(),,((),( vuvuvuvu FIT  =  where ),( vu T   = Max(S(P1)), ) , (vu I   = Max(S(P2)), ), (v u F   = Min(S(P3)) Definition:2.5 Let G be a bijective neurtosophic graph. An edge of G is said to T-bridge if ),( vu T   < ),( 'vu T   where ) ,( 'vu T   is the connectedness between u and v by removing any edge. An edge of G is said to I-bridge if ),( v u I   < ),( 'vu I   where ),( 'vu I   is the connectedness between u and v by removing any edge. An edge of G is said to F-bridge if ), (v u F   > ),( 'vu F   where ),( 'vu F   is the connectedness between u and v by removing any edge. Definition: 2.6 An edge of G is said to be a neutrosophic bridge if it is Tbridge, I-bridge and Fbridge. 3. An Algorithm for finding the bridges of any bijective neutrosophic graph Input: A bijective neutrosophic graph G = (V, E, T, I, F), where each edge has associated truth-membership T, indeterminacy-membership I, and falsity-membership F. Output: Classification of each bridge as T-Bridge, I-Bridge, or F-Bridge. Step 1: Begin with a bijective neutrosophic graph G. Select an arbitrary crisp cycle C* consisting of nedges, where n ≥ 3 Step 2: If Tbridge or Ibridge of G are required, then identify an edge T  = T n i  1=  or I  = I n i  1=  by considering all the edges of C* respectively. __________________________________________________________________________________________________ D.Rajalaxmi, V.Vijaya and S.Revathi, An Algorithmic Approach to Bridge Detection in Bijective Neutrosophic Graph with Application in Cancer Diagnosis Neutrosophic Sets and Systems, Vol. 96, 2026 288 Step 3: Remove the identified edge η from G. Step 4: Choose another cycle C* in G with any number of edges and repeat step 2 & step 3 until no such cycle remains in G. Step 5: After removal of all η’s from G, the resulting graph comprises the Tbridges or I – bridges of G respectively. Step 6: If Fbridges is required for the chosen graph then find F  = F i n V  1 − by considering all the edges of C*. Step 7: Repeat step 3 and step 4. Step 8: Upon removal of all F  ’s from the bijective neutrosophic graph G, the resulting graph comprises the Fbridges of G. The Tbridges and Ibridges are the strongest connections between the vertices in the graph G and F – bridges is the weakest connections between the vertices in the graph G. The resulting graph which is obtained is the Tmaximum spanning sub graph or Imaximum spanning sub graph of G. Also one can obtain the F-minimum spanning subgraph of G. Example:3.1 Figure 2:Bijective neutrosophic graph Let us find the Tbridges of the above figure 2 by Considering the cycle C1* = v1,,v3,v4,v5,v1 of length 4. Here η1= min(0.13, 0.15, 0.80, 0.59) η1= 0.13 __________________________________________________________________________________________________ D.Rajalaxmi, V.Vijaya and S.Revathi, An Algorithmic Approach to Bridge Detection in Bijective Neutrosophic Graph with Application in Cancer Diagnosis Neutrosophic Sets and Systems, Vol. 96, 2026 289 Figure 3 In the above figure 3 edge e7 with 0.13 truth membership value has to be removed from G. Let the next cycle be C2* = v1,v4,v5,v1 of length 3. Here η2= min(0.55, 0.80, 0.59) η2= 0.55 Figure 4 Clearly in the above figure 4 the edge e4 with 0.55 truth membership value has to be removed from G. So Let the next cycle be C3* = v1,v2,v3,v4,v5,v1 of length 5. Here η3= min(0.58, 0.33, 0.15, 0.80, 0.59) η3= 0.15 __________________________________________________________________________________________________ D.Rajalaxmi, V.Vijaya and S.Revathi, An Algorithmic Approach to Bridge Detection in Bijective Neutrosophic Graph with Application in Cancer Diagnosis Neutrosophic Sets and Systems, Vol. 96, 2026 290 Figure 5 Clearly in the above figure 5 the edge e3 with 0.15 as truth membership value has to be removed from G. Since no cycles remains after the removal of edges, the following resulting graph represents all the T-Bridges of G Figure 6: T-Bridges of G Hence the TBridges of G are (v1,v2) , (v2,v3) , (v4,v5) and (v5,v1) If the above algorithm is applied for Indeterminancy then the IBridge of G can be obtained. Hence the IBridges of G are (v1,v3) , (v1,v5) , (v2,v3) and (v3,v4) Similarly F-Bridges of G are (v1,v5) , (v1,v4) , (v1,v2) and (v2,v3). In some practical situation if it is necessary to consider all the membership functions in the same time then we can use score function to find the bridges of bijective neutrosophic graph. Definition:[12] The score function is defined as S(u) = 3 )()()( uuu FIT  ++ The same algorithm can be used to find the bridges of the bijective neutrosophic graph after find the score function for all the vertices and edges. __________________________________________________________________________________________________ D.Rajalaxmi, V.Vijaya and S.Revathi, An Algorithmic Approach to Bridge Detection in Bijective Neutrosophic Graph with Application in Cancer Diagnosis Neutrosophic Sets and Systems, Vol. 96, 2026 291 Example: Now let us find the bridges of the graph given in figure after finding the score function S(v1) = 3 )()()( 111 vvv FIT  ++ = 3 30.071.060.0 ++ = 0.54 S(v2) = 0.56, S(v3) = 0.58 and so on. Figure 7 For finding the bridges of G let us first consider a cycle C1* = v1,v2,v3,v4,v5,v1 of length 5. Figure 8 Clearly the edge e2 is to be removed as it has minimum value among the other edge values in the considered cycle. Now let us choose another cycle C2* = v1,v3,v4,v5,v1 of length 4. __________________________________________________________________________________________________ D.Rajalaxmi, V.Vijaya and S.Revathi, An Algorithmic Approach to Bridge Detection in Bijective Neutrosophic Graph with Application in Cancer Diagnosis Neutrosophic Sets and Systems, Vol. 96, 2026 292 Figure 9 The above marked edge e3 is the next edge with minimum score value which has to be removed next. Now consider another cycle C3* = v1,v4,v5,v1 of length 3. Figure 10 Clearly if the above marked edge is removed, then the bridges of G are (v1,v2) , (v1,v3) , (v4,v5) and (v5,v1). Note: The score function is one of the methods of de neutrosophication. So score function need not be bijective. 4. Applications Now, Let’s use the above discussed algorithm to find a better diagnosis for cancer. Let’s consider a specific case in which we have a patient with the more or less equal symptoms of Inflammation of Gastrointestinal Tract (GT), Chronic Cough(C) and Fatigue(F).i.e. As per the data, he exhibits 48% of the typical symptoms associated with Inflammation of Gastrointestinal Tract, 40% clinical symptoms for chronic cough and 42% of Fatigue. And we suspect that the patient might be suffering from Colorectal Cancer(CRC), Lung Cancer(LC), Tuberculosis(TB) and Inflammatory Bowel Disease(IBD).i.e. There is approximately equal chance for him to suffer from all the above mentioned disease. In __________________________________________________________________________________________________ D.Rajalaxmi, V.Vijaya and S.Revathi, An Algorithmic Approach to Bridge Detection in Bijective Neutrosophic Graph with Application in Cancer Diagnosis