Graph-theoretic analysis demonstrates structural bounds and algorithms for maximum geodesic edge-covering numbers across graph classes, revealing links to graph domination.
A graph G is k-geodetic edge traceable if every edge of G lies in at least one geodesic of length k. The maximum geodesic edge-covering number gecₘₐₓ(G) is the minimum number of largest fixed-length geodesics that cover all edges of G. We study some properties of k-geodetic edge traceable graphs and establish relations between gecₘₐₓ(G) and the domination number of a connected graph. We investigate the k-geodetic edge traceability of the complete bipartite graph Km,n , and provide an algorithm to compute gecₘₐₓ(Kn,m) . We also study the k-geodetic edge traceability of trees and product graphs, namely the Cartesian, strong, lexicographic, and Corona products, and obtain bounds for gecₘₐₓ of these product graphs.
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Guragain et al. (2026) studied this question.
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