On the Forcing Domination and the Forcing Total Domination Numbers of a Graph

Abstract

Let G be a connected graph with at least two vertices and S a $\gamma _{t}$-set of G. A subset $T \subseteq S$ is called a forcing subset for S if S is the unique $\gamma _{t}$-set containing T. The forcing total domination number of S, denoted by $f_{\gamma _{t}}(S)$, is the cardinality of a minimum forcing subset of S. The forcing total domination number of G, denoted by $f_{\gamma _{t}}(G)$ is defined by $f_{\gamma _{t}}(G)$ = min $\lbrace f_{\gamma _{t}}(S)\rbrace$, where the minimum is taken over all minimum total dominating sets S in G. Some general properties satisfied by this concepts are studied. The forcing total dominating number of certain standard graphs are determined. It is shown that for every pair a, b of integers with $0 \le a < b$ and $b \ge 1$, there exists a connected graph G such that $f_{\gamma _{t}}(G) = a$ and $\gamma _{t}(G) = b$, where $\gamma _{t}(G)$ is total domination number of G. It is also shown that for every pair a,b of integers with $a \ge 0$ and $b \ge 0$, there exists a connected graph G such that $f_{{\gamma }_{t}}(G) = a$ and $f_{\gamma }(G) = b$, where $f_{\gamma }(G)$ is the forcing domination number of G.