L-Hop Independent Sequences in Graphs

Abstract

Let $G$ be a graph. A sequence of distinct vertices $Q=(a_1, a_2, \ldots, a_n)$ of $G$ is called an L-hop independent sequence if $n=1$ or if $d_G(a_i,a_j)\neq 2$ for each $i\neq j$, where $i,j\in\{1,2,\ldots, n\}$ and $ N_G[a_s] \backslash \displaystyle \bigcup_{t=1}^{s-1} N_G(a_t) \neq \varnothing$ for each $s\in \{2,\ldots,n\}$. The L-hop independence number of $G$, denoted by $\alpha_{Lh}(G)$, 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.