A graph $G$ of order $n$ is arbitrarily partitionable~(AP in abbreviation)~if for each sequence $\tau=(n_{1},n_{2},\ldots,n_{k})$ of positive integers such that it sums up to $n$,~there exists a partition $(V_{1},\ldots,V_{k})$ of the vertex set $V$ such that $|V_{i}|=n_{i}$ and $V_{i}$ induces a connected subgraph for all $i=1,\ldots,k$.~A star-like tree~$S(a,b,c)$ is a tree homeomorphic to the star $K_{1,3}$.~In this paper,~we focus on the arbitrarily partition of the line graph of $S(a,b,c)$.~We give the necessary and sufficient conditions for graphs $L(S(2,b,c))$ and $L(S(3,b,c))$ being AP.~Furthermore,~we provide three sufficient conditions for which $L(S(a,b,c))$ is not AP.