Some classes of dispersible dcsl-graphs

A distance compatible set labeling (dcsl) of a connected graph $G$ is an injective set assignment $f : V(G) \rightarrow 2^{X},$ $X$ being a non empty ground set, such that the corresponding induced function $f^{\oplus} :E(G) \rightarrow 2^{X}\setminus \{\phi\}$ given by $f^{\oplus}(uv)= f(u)\oplus f(v)$ satisfies $ |f^{\oplus}(uv)| = k_{(u,v)}^{f}d_{G}(u,v) $ for every pair of distinct vertices $u, v \in V(G),$ where $d_{G}(u,v)$ denotes the path distance between $u$ and $v$ and $k_{(u,v)}^{f}$ is a constant, not necessarily an integer, depending on the pair of vertices $u,v$ chosen. $G$ is distance compatible set labeled (dcsl) graph if it admits a dcsl. A dcsl $f$ of a $(p, q)$-graph $G$ is dispersive if the constants of proportionality $k^f_{(u,v)}$ with respect to $f, u \neq v, u, v \in V(G)$ are all distinct and $G$ is dispersible if it admits a dispersive dcsl. In this paper we proved that all paths and graphs with diameter less than or equal to $2$ are dispersible.