Chordal bipartite graphs of bounded tree- and clique-width*1

Abstract

A bipartite graph is chordal bipartite if every cycle of length at least six has a chord. In the class of chordal bipartite graphs the tree-width and the clique-width are unbounded. Our main results are that chordal bipartite graphs of bounded vertex degree have bounded tree-width and that k -fork-free chordal bipartite graphs have bounded clique-width, where a k -fork is the graph arising from a K 1, k +1 by subdividing one edge once. (Note that a bipartite graph has vertex degree at most k if and only if it is K 1, k +1 -free.) This implies polynomial-time solvability for a variety of algorithmical problems for these graphs.