Definitions for

**"graph G"**a finite set, V(G) of objects, called "vertices", together with a set, E(G), of unordered pairs of distict vertices

a k-leaf power, fixed k if there is a tree T whose leaves correspond to the vertices of G in such a way that two vertices are adjacent in G precisely when their distance in T is at most k

a k -tree if it is K k or is obtained from a k -tree by adding a new vertex adjacent to a k -clique

a Helly circle graph if G is a circle graph and there exists a model of G by chords such that every three pairwise intersecting chords intersect at the same point

an e-circle graph if and only if it is a circle graph

an e-circle graph if we can find a model of straight lines on the cartesian plane according to the following procedure

a divisor graph if there exists a set S of positive integers such that G is isomorphic to G(S)

a path tolerance graph (PT-graph) iff G is a PT(t)-graph for some tolerance t

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**Keywords:**extremal

a extremal graph if and only