The Upper and The Forcing Geodetic Hop Domination Numbers of a Graph

In a connected graph G, a minimal geodetic hop dominating set S is defined as a set where no proper subset of S can serve as a geodetic hop dominating set of G. The upper geodetic hop domination number, denoted as \(\gamma _{hg}^{+}(G)\) represents the largest possible size of a minimal geodetic hop dominating set within graph G, exploring fundamental properties inherent to this concept. The concept of upper geodetic hop domination numbers in graph theory provides valuable insights into the structure and behaviour of connected graphs. For any given positive integers a and b, where 2 is less than or equal to a and less than or equal to b, there exists a connected graph G such that its geodetic hop domination number, denoted as \({\gamma _{hg}}(G)\) , equals a, while its upper geodetic hop domination number \(\gamma _{hg}^{+}(G)\) equals b. In this context, a subset T of a \(\gamma _{hg}\) -set S in G is termed a forcing subset if S is the unique \(\gamma _{hg}\) -set containing T. The cardinality of the minimum forcing subset, denoted as \(f_{\gamma _{hg}}(S)\) , represents the forcing geodetic hop domination number of S. The minimum forcing geodetic hop domination number of G, \(f_{\gamma _{hg}}(G)\) , is then defined as the minimum \(f_{\gamma _{hg}}(S)\) taken over all \(\gamma _{hg}\) -sets of G. It has been shown that for any pair of positive integers a and b, where a is greater than or equal to 0, and b is greater than \(a + 3\) , there exists a connected graph G satisfying \(f_{\gamma _{hg}}(G) = a\) and \(\gamma _{hg}(G) = b\) . These findings underscore the rich diversity and complexity inherent in graph structures, shedding light on fundamental properties that govern their behaviour.