Legal Closed Hop Neighborhood Independent Sequences in Graphs (European Journal of Pure and Applied Mathematics, 2025)
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: [$(i)$] $N_G^2[x_i] \setminus \bigcup_{j=1}^{i-1} N_G^2[x_j] eq \empty$ for each $i \in \{2, 3, ..., k\}$, and [$(ii)$] $d_G (x_s, x_t) eq 1$ for each $s,t \in \{1, 2, ..., k\}$, where $s eq t$. 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 $θ(G)$. 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.
Keywords: independent set legal closed hop neighborhood independent sequence, legal closed hop neighborhood independence number