# Minimally triangle-saturated graphs: adjoining a single vertex

MacDougall, James A. and Eggleton, Roger B. (2002) Minimally triangle-saturated graphs: adjoining a single vertex. Australasian Journal of Combinatorics, 25 . pp. 263-278.

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A graph $G$ is triangle-saturated if every possible edge addition to $G$ creates one or more new triangles (3-cycles). Such a graph is minimally triangle-saturated if removal of any edge from $G$ leaves a graph that is not triangle-saturated. This paper investigates adding a single new vertex to a triangle-saturated graph so as to produce a new triangle-saturated graph, and determines the conditions under which the extended graph is minimally saturated. Particular attention is given to minimally saturated extensions which are {\em primitive} (no two vertices have the same neighbourhood). The results are applied to construct primitive maximal triangle-free graphs of every order $n \geq 9$.