Hamiltonian laceability in the shadow distance graph of path graphs

Abstract

A connected graph $G$ is termed hamiltonian-t-laceable $\left(t^{\star}\right.$-laceable) if there exists in it a hamiltonian path between every pair (at least one pair) of distinct vertices $u$ and $v$ with the property $d(u, v)=t, 1 \leq t \leq \operatorname{diam}(G)$, where $t$ is a positive integer. In this paper, we establish laceability properties in the edge tolerant shadow distance graph of the path graph $P_n$ with distance set $D_s=\{1,2 k\}$.