The Domination Number of a Graph \(P_k ((k_1, k_2), (k_3, k_4))\)

Monthiya Ruangnai, Sayan Panma


For each \(k, k_1, k_2, k_3, k_4 \in \mathbb{N}\), we will denote by \(P_k \big((k_1, k_2), (k_3, k_4)\big)\) a tree of \(k+k_1+k_2+k_3+k_4+1\) vertices with the degree sequence \((1,1,1,1,2,2,2,\dots,2,3,3)\). Let \(\alpha_{k_1}, \beta_{k_2}, \sigma_{k_3}\), and \(\delta_{k_4}\) 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 \(P_k \big((k_1, k_2), (k_3, k_4)\big)\) is a dominating set of \(P_k \big((k_1, k_2), (k_3, k_4)\big)\) if every vertex in \(V\big(P_k \big((k_1, k_2), (k_3, k_4)\big)\big)-S\) is adjacent to some vertex in \(S\). We investigate the dominating set of minimum cardinality of a graph \(P_k \big((k_1, k_2), (k_3, k_4)\big)\) to obtain the domination number of this graph. Finally, we determine an upper bound on the domination number of a graph \(P_k \big((k_1, k_2), (k_3, k_4)\big)\).


Domination number; Tree; A dominating set of a graph; The domination number of a graph; The domination number of a tree

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