Let \( G \) be a simple graph, and let \( p \) be a positive integer. A subset \( D \subseteq V(G) \) is a \( p \)-dominating set of the graph \( G \) if every vertex \( v \in V(G) – D \) is adjacent to at least \( p \) vertices of \( D \). The \( p \)-domination number \( \gamma_p(G) \) is the minimum cardinality among the \( p \)-dominating sets of \( G \). Note that the \( 1 \)-domination number \( \gamma_1(G) \) is the usual domination number \( \gamma(G) \).
A subset \( S \subseteq V(G) \) is said to be a total dominating set if every vertex in \( V(G) \) has at least one neighbor in \( S \), and it is a connected dominating set if the graph induced by \( S \) is connected. The total domination number \( \gamma_t(G) \) represents the cardinality of a minimum total dominating set of \( G \) and the connected domination number \( \gamma_c(G) \) the cardinality of a minimum connected dominating set.
Fink and Jacobson showed in 1985 that if \( G \) is a graph with \( \Delta(G) \geq p \geq 2 \), then \(\gamma_p(G) \geq \gamma(G) + p – 2.\)
In this paper, we will give some sufficient conditions for a graph \( G \) such that \(\gamma_p(G) \geq \gamma(G) + p – 1.\)
We will show that for block graphs \( G \) the inequality \(\gamma_p(G) \geq \gamma_t(G) + p – 2 \) is valid and that for trees \( T \) the inequality \(\gamma_p(T) \geq \gamma_c(T) + p – 1\) holds. Further, we characterize the trees \( T \) with \(\gamma_p(T) = \gamma_c(T) + p – 1,\) \(\gamma_p(T) = \gamma_t(T) + p – 2, \gamma_p(T) = \gamma_t(T) + p – 1,\) and \(\gamma_p(T) = \gamma(T) + p – 1.\)