Graph Extremities Defined by Search Algorithms

Graph search algorithms have exploited and in some cases identi ed graph extremities, such as the leaves of a tree and the simplicial vertices of a chordal graph. Recently, several well-known graph search algorithms have been collectively expressed as two generic algorithms called MLS and MLSM, each of which instantiates with parameters de ning the set of labels to be used, a partial order on labels used to choose at each step a vertex of maximal label and the way a label is initialized and incremented. In this paper, we investigate the properties of the vertex that is numbered 1 by MLS on a chordal graph and by MLSM on an arbitrary graph. We show that the minimal separators included in the neighborhood of this vertex are totally ordered by inclusion and that this vertex is indeed an extremity, as it belongs to a moplex of the graph. When the order on labels is total, this extremity property is still stronger since all the vertices of this moplex are numbered consecutively. When MLS is run on a non-chordal graph with a total order on labels, vertex number 1 is a weaker kind extremity, called an OCF-vertex.