The NLC-width and clique-width for powers of graphs of bounded tree-width

The kk-power graph   of a graph GG is a graph with the same vertex set as GG, in that two vertices are adjacent if and only if, there is a path between them in GG of length at most kk. A kk-tree-power graph   is the kk-power graph of a tree, a kk-leaf-power graph   is the subgraph of some kk-tree-power graph induced by the leaves of the tree.We show that (1) every kk-tree-power graph has NLC-width at most k+2k+2 and clique-width at most k+2+max{⌊k2⌋−1,0}, (2) every kk-leaf-power graph has NLC-width at most kk and clique-width at most k+max{⌊k2⌋−2,0}, and (3) every kk-power graph of a graph of tree-width ll has NLC-width at most (k+1)l+1−1(k+1)l+1−1, and clique-width at most 2⋅(k+1)l+1−22⋅(k+1)l+1−2.