Legal Closed Hop Neighborhood Independent Sequences in Graphs

Abstract

Let $G$ be a graph. A sequence $Q=(x_1, x_2,..., x_k)$ of distinct vertices of $G$ is called a legal closed hop neighborhood independent sequence (LCHNI sequence) if it is satisfies the following two conditions: [$(i)$] $N_G^2[x_i] \setminus \bigcup_{j=1}^{i-1} N_G^2[x_j] \neq \empty$ for each $i \in \{2, 3, ..., k\}$, and [$(ii)$] $d_G (x_s, x_t) \neq 1$ for each $s,t \in \{1, 2, ..., k\}$, where $s \neq t$. The legal closed hop neighborhood independence number (LCHNI number) of $G$ is the maximum length of an LCHNI sequence of $G$, and this is denoted by $\theta(G)$. In this paper, the authors initiate the study of legal closed hop neighborhood independent sequence of a graph and give characterization to some special graphs, shadow graphs, and the join of two graphs. Subsequently, the authors determine the corresponding legal closed neighborhood independence numbers of these graphs.