Further Results on the Hop Domination Number of a Graph

A hop dominating set S in a connected graph G is called a minimal hop dominating set if no proper subset of S is a hop dominating set of G. The upper hop domination number gamma(+)(h) (G) of G is the maximum cardinality of a minimal hop dominating set of G. Some general properties satisfied by this concept are studied. It is shown that for every two positive integers a and b where 2 <= a <= b, there exists a connected graph G such that gamma(h)(G) = a and gamma(+)(h) (G) = b. It is proved that minimal hop dominating set is NP-complete. It is proved that gamma(h)(G) and gamma(G) are in general incomparable. It is shown that for every pair of positive integers a and b with a >= 2 and b >= 1, there exists a connected graph G such that gamma(h)(G) = a and gamma(G) = b. We present an algorithm to compute minimal hop dominating set of G. Finally, we formulate an Integer linear programming problem to compute the hop domination number of G.