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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 $$\\mathcal{A}(G)$$A(G) of a chordal graph G is a directed graph that represents the intersections between maximal cliques of G by nodes and the inclusion relation 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 $$k \\ge 3$$kź3. The class $$\\mathcal{L}_k$$Lk of k-leaf powers consists of graphs $$G=(V,E)$$G=(V,E) that have a k-leaf root, that is, a tree T with leaf set V, where $$xy \\in E$$xyźE if and only if the T-distance between x and y is at most k. Structural characterizations 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 $$\\mathcal{L}= \\bigcup _{k=2}^\\infty \\mathcal{L}_k$$L=źk=2źLk, 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 $$\\mathcal{L}$$L is contained in that class.

Towards a Characterization of Leaf Powers by Clique Arrangements