As a by-product, it is proved that the class of probe block graphs is the intersection between the classes of chordal graphs
and probe diamond-free graphs.
In particular, kB([GAMMA]) is hereditary if and only if [GAMMA] is a chordal graph
The notion of signed-eliminable graphs is a generalization of chordal graphs
to signed graphs, and the above result is also a signed-graphic generalization of Stanley's well-known result (Stanley (1972)) on a characterization of free graphic arrangements in terms of chordal graphs
Lemma 4.11 Let G be a chordal graph
and let [psi] be an LBFS ordering of G.
The following theorem of  gives a necessary and sufficient condition for a vertex to be last in a Maximal Neighbourhood Search (MNS) in a chordal graph
It was also shown by Jacobson and Peters  that [GAMMA](G) = [[beta].sub.0](G) for any chordal graph
G, and so [GAMMA](G) can be computed for chordal graphs
in polynomial time.
From [Dir61] we know that every chordal graph
has a simplicial vertex, i.e, a vertex whose neighbors induce a clique.
Recall that a chordal graph
is a graph with no induced cycles of length at least four [26, 51].
But the problem remains NP-complete for chordal graphs
, undirected path graphs, split graphs, tripartite graphs, graphs that are the complement of a bipartite graph , and planar graphs if the weights are of arbitrary sign .
. If the graph g is chordal, then the matrix [N.sup.T]DN is also chordal when D is diagonal.
In this article we prove that the problem of computing the minimum cardinality of an open 0-monopoly in a graph is NP-complete even restricted to bipartite or chordal graphs
. In addition we present some general bounds for the minimum cardinality of open k-monopolies and we derive some exact values.