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.
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Nevries, R., Rosenke, C. (2015). Towards a Characterization of Leaf Powers by Clique Arrangements. In: Italiano, G.F., Margaria-Steffen, T., Pokorný, J., Quisquater, JJ., Wattenhofer, R. (eds) SOFSEM 2015: Theory and Practice of Computer Science. SOFSEM 2015. Lecture Notes in Computer Science, vol 8939. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46078-8_30
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DOI: https://doi.org/10.1007/978-3-662-46078-8_30
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