The {2}-domination in rooted product graphs

<p>A <span class="math inline">\(\{2\}\)</span>-dominating function (<span class="math inline">\(\{2 \}\)</span>DF) on a graph <span class="math inline">\(G=(V(G),E(G))\)</span> is a function <span class="math inline">\(f : V(G) \rightarrow \{0,1,2 \}\)</span> such that <span class="math inline">\(f(N[v]) \geq 2\)</span> for every <span class="math inline">\(v \in V(G)\)</span>, where <span class="math inline">\(N[v]\)</span> is the closed neighourhood of <span class="math inline">\(v\)</span>. The <span class="math inline">\(\{2\}\)</span>-domination number of <span class="math inline">\(G\)</span> is the minimum weight <span class="math inline">\(\omega(f) = \sum\limits_{v \in V(G)} f(v)\)</span> among all <span class="math inline">\(\{2 \}\)</span>-dominating functions on <span class="math inline">\(G\)</span>. In this article, we prove that if <span class="math inline">\(G\)</span> and <span class="math inline">\(H\)</span> are graphs with no isolated vertex, then for any vertex <span class="math inline">\(v \in V(H)\)</span> there are six closed formulas for the <span class="math inline">\(\{2\}\)</span>-domination number of the rooted product graph <span class="math inline">\(G \circ_v H\)</span>. We also characterize the graph <span class="math inline">\(G\)</span> and <span class="math inline">\(H\)</span> that satisfy each of these formulas.</p>