Further results on total κ-subdominating functions in trees

A two-valued function f defined on the vertices of a graph G = (V, E), f: V → {-1,1} is an opinion function. For each vertex v of G, the vote of v is the sum of the function values of f over the open neighborhood of v. A total k-subdominating function (TkSF) of a graph G is an opinion function for which at least k of the vertices have a vote value of at least one. The total k-subdomination number, γk^t_s(G), of G is the minimum value of f(V) over all TkSFs of G where f(V) denotes the sum of the function values assigned to the vertices under f. Trees of even order n achieving the maximum possible total k-subdomination number (namely, 2k - n) when n/2 ≤ k ≤ n/2 + 3 were characterized in [9]. Our aim in this paper is to characterize those trees of order n achieving the maximum possible total k-subdomination number (namely, 2k - n) when n is odd and (n + 3)/2 ≤ k ≤ (n + 5)/2.