On 1-Near Mean Cordial Labeling of the Strong Product of Two Paths: Characterization and a Verification Algorithm

The 1-Near mean cordial labeling is a variation of mean cordial labeling. Let G=V,E be a simple graph. A surjective function h:VG⟶0,1,2 satisfies a 1-Near mean cordial labeling if for each edge xy, the induced map:h∗xy=0;if hx+hy/2 is an integer1;otherwise fulfills the constraint E0−E1≤1, where E0 and E1 are the number of edges with labels zero and one, respectively. If the graph G accepts a 1-Near mean cordial labeling, it is called a 1-Near mean cordial graph. This paper explores 1-Near mean cordial labeling for the strong product of two path graphs (denoted Pw⊠Pz). Furthermore, we present an O w×z time algorithm to check 1-Near mean cordiality for the strong product of two paths. Theoretical results demonstrate that Pw⊠Pz is 1-Near mean cordial for all w,z≥2. These findings extend the scope of 1-Near mean cordial labeling to strong product graphs and provide efficient computational tools for its verification.