2015 | OriginalPaper | Chapter
Towards a Characterization of Leaf Powers by Clique Arrangements
Authors : Ragnar Nevries, Christian Rosenke
Published in: SOFSEM 2015: Theory and Practice of Computer Science
Publisher: Springer Berlin Heidelberg
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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
${{\cal A}}(G)$
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.