Some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1

Let $G=(V,E)$ be a simple graph with vertex set $V$ and edge set $E$, and let $f$ be a function $f:V\mapsto \{0,1,2\}$. A vertex $u$ with $f(u)=0$ is said to be undefended with respect to $f$ if it is not adjacent to a vertex with positive weight. The function $f$ is a weak Roman dominating function (WRDF) if each vertex $u$ with $f(u)=0$ is adjacent to a vertex $v$ with $f(v)>0$ such that the function $f_u:V\mapsto \{0,1,2\}$, defined by $f_u(u)=1$, $f_u(v)=f(v)-1$ and $f_u(w)=f(w)$ if $w\in V-\{u,v\}$, has no undefended vertex. The weight of $f$ is $w(f)=\sum _{v\in V}f(v)$. The weak Roman domination number, denoted ${\gamma }_r(G)$, is the minimum weight of a WRDF in G. The domination number, denoted $\gamma (G)$, is the minimum cardinality of a dominating set in $G$. In this paper, we give some sufficient conditions for a tree to have its weak Roman domination number be equal to its domination number plus 1 ( ${\gamma }_r(T)=\gamma (T)+1$) by recursion and construction.