Towards a Characterization of Leaf Powers by Clique Arrangements

Abstract

In this paper, we use the new notion of clique arrangements to suggest that leaf powers are a natural special case of strongly chordal graphs. The clique arrangement ${{\cal A}}(G)$ of a chordal graph G is a directed graph that represents the intersections between maximal cliques of G by nodes and the mutual inclusion of these vertex subsets by arcs. Recently, strongly chordal graphs have been characterized as the graphs that have a clique arrangement without bad k-cycles for k ≥ 3.

The class ${{\cal L}}_k$ of k-leaf powers consists of graphs G = (V,E) that have a k-leaf root, that is, a tree T with leaf set V, where xy ∈ E if and only if the T-distance between x and y is at most k. Structure and linear time recognition algorithms have been found for 2-, 3-, 4-, and, to some extent, 5-leaf powers, and it is known that the union of all k-leaf powers, that is, the graph class ${{\cal L}} = \bigcup_{k=2}^\infty {{\cal L}}_k$, forms a proper subclass of strongly chordal graphs. Despite that, no essential progress has been made lately.

In this paper, we characterize the subclass of strongly chordal graphs that have a clique arrangement without certain bad 2-cycles and show that ${{\cal L}}$ is contained in that class.