The Domination Number of a Graph \(P_k ((k_1, k_2), (k_3, k_4))\) - Ruangnai, Monthiya; Panma, Sayan (Communications in Mathematics and Applications, 2019)
For each \(k, k_1, k_2, k_3, k_4 N\), we will denote by \(P_k ((k_1, k_2), (k_3, k_4))\) a tree of \(k+k_1+k_2+k_3+k_4+1\) vertices with the degree sequence \((1,1,1,1,2,2,2,…,2,3,3)\). Let \(α_k_1, β_k_2, σ_k_3\), and \(δ_k_4\) be all four endpoints of the graph. Let the distance between both vertices of degree 3 be equal to \(k\). A subset \(S\) of vertices of a graph \(P_k ((k_1, k_2), (k_3, k_4))\) is a dominating set of \(P_k ((k_1, k_2), (k_3, k_4))\) if every vertex in \(V(P_k ((k_1, k_2), (k_3, k_4)))-S\) is adjacent to some vertex in \(S\). We investigate the dominating set of minimum cardinality of a graph \(P_k ((k_1, k_2), (k_3, k_4))\) to obtain the domination number of this graph. Finally, we determine an upper bound on the domination number of a graph \(P_k ((k_1, k_2), (k_3, k_4))\).