The Clique-Width of Tree-Power and Leaf-Power Graphs

Abstract

The k-power graph of a graph G is the graph in which two vertices are adjacent if and only if there is a path between them in G of length at most k. We show that (1.) the k-power graph of a tree has NLC-width at most k + 2 and clique-width at most $k+2+\max(\lfloor \frac{k}{2}\rfloor -1,0)$, (2.) the k-leaf-power graph of a tree has NLC-width at most k and clique-width at most $k+ \max(\lfloor \frac{k}{2}\rfloor -2,0)$, and (3.) the k-power graph of a graph of tree-width l has NLC-width at most (k + 1)^l + 1− 1 and clique-width at most 2·(k + 1)^l + 1− 2.