In a finite simple undirected graph, a vertex is simplicial if its neighborhood is a clique. We say that, for
k ≥ 2, a graph
G = (
V
G
,
E
G
) is the
k-simplicial power of a graph
H = (
V
H
,
E
H
) (
H a
root graph of
G) if
V
G
is the set of all simplicial vertices of
H, and for all distinct vertices
x and
y in
V
G
,
xy ∈
E
G
if and only if the distance in
H between
x and
y is at most
k. This concept generalizes
k-leaf powers introduced by Nishimura, Ragde and Thilikos which were motivated by the search for underlying phylogenetic trees;
k-leaf powers are the
k-simplicial powers of trees. Recently, a lot of work has been done on
k-leaf powers and their roots as well as on their variants phylogenetic roots and Steiner roots. For
k ∈ {3,4,5},
k-leaf powers can be recognized in linear time, and for
k ∈ {3,4}, structural characterizations are known. For all other
k, recognition and structural characterization of
k-leaf powers is open.
Since trees and block graphs (i.e., connected graphs whose blocks are cliques) have very similar metric properties, it is
natural to study k-simplicial powers of block graphs. We show that leaf powers of trees and simplicial powers of block graphs are closely related,
and we study simplicial powers of other graph classes containing all trees such as ptolemaic graphs and strongly chordal graphs.
Keywords Graph powers - leaf powers - simplicial powers - forbidden induced subgraph characterization - chordal graphs - block graphs - ptolemaic graphs - strongly chordal graphs