L-Hop Independent Sequences in Graphs (European Journal of Pure and Applied Mathematics, 2025)
Let $G$ be a graph. A sequence of distinct vertices $Q=(a_1, a_2, …, a_n)$ of $G$ is called an L-hop independent sequence if $n=1$ or if $d_G(a_i,a_j) eq 2$ for each $i eq j$, where $i,j\in\{1,2,…, n\}$ and $ N_G[a_s] \backslash \displaystyle \bigcup_{t=1}^{s-1} N_G(a_t) eq \varnothing$ for each $s\in \{2,…,n\}$. The L-hop independence number of $G$, denoted by $α_{Lh}(G)$, is the maximum length among all L-hop independent sequences in $G$. This study explores and characterizes the L-hop independent sequences in some graphs, complementary prism of any graph, and in the join of two graphs. Some formulas and bounds of L-hop independence number with respect to the order of a graph and other parameters in graph theory are derived. Moreover, some relationships of L-hop independence with hop independence and legal hop independence are established.
Keywords: L-sequence, clique L-sequence, clique L-Grundy dominating sequence, L-hop independent sequence; L-independence number