Secure domination parameters of Halin graph with perfect k-ary tree

Secure domination parameters of Halin graph with perfect k-ary tree

Let G be a simple undirected graph. A planar graph known as a Halin graph(HG) is characterised by having three connected and pendent vertices of a tree that are connected by an outer cycle. A subset $S$ of $V$ is said to be a dominating set of the graph $G$ if each vertex $u$ that is part of $V$ is dominated by at least one element $v$ that is a part of $S$. The domination number of a graph is denoted by the $\gamma(G)$, and it corresponds to the minimum size of a dominating set. A dominating set $S$ is called a secure dominating set if for each $v\in V \backslash S$ there exists $u\in S$ such that $v$ is adjacent to $u$ and $S_1 = (S \backslash \{v\})\cup \{u\} $ is a dominating set. The minimum cardinality of a secure dominating set of $G$ is equal to the secure domination number $\gamma_s (G)$. In this article we found the secure domination number of Halin graph(HG) with perfet k-ary tree and also we determined secure domination of rooted product of special trees.

Let G be a simple undirected graph. A planar graph known as a Halin graph(HG) is characterised by having three connected and pendent vertices of a tree that are connected by an outer cycle. A subset $S$ of $V$ is said to be a dominating set of the graph $G$ if each vertex $u$ that is part of $V$ is dominated by at least one element $v$ that is a part of $S$. The domination number of a graph is denoted by the $\gamma(G)$, and it corresponds to the minimum size of a dominating set. A dominating set $S$ is called a secure dominating set if for each $v\in V \backslash S$ there exists $u\in S$ such that $v$ is adjacent to $u$ and $S_1 = (S \backslash \{v\})\cup \{u\} $ is a dominating set. The minimum cardinality of a secure dominating set of $G$ is equal to the secure domination number $\gamma_s (G)$. In this article we found the secure domination number of Halin graph(HG) with perfet k-ary tree and also we determined secure domination of rooted product of special trees.