No abstract

Suppose [Formula: see text] is a subgraph of a convex complete graph [Formula: see text] and [Formula: see text] contains no boundary edge of [Formula: see text] and [Formula: see text]. We determine necessary and sufficient conditions on [Formula: see text] such that [Formula: see text] admits a triangulation. For [Formula: see text], we investigate the possibility of placing [Formula: see text] in [Formula: see text] such that [Formula: see text] admits a triangulation for certain families of graphs [Formula: see text]. These results are then applied to determine the convex skewness of the convex graphs of the form [Formula: see text].

The restricted triangulation existence problem on a given graph decides whether there exists a triangulation on the graph’s vertex set that is restricted with respect to its edge set. Let G = C(n, S) be a circulant graph on n vertices with jump value set S. We consider the restricted triangulation existence problem for G. We determine necessary and sufficient conditions on S for which G admitting a restricted triangulation. We characterize a set of jump values S(n) that has the smallest cardinality with C(n, S(n)) admits a restricted triangulation. We present the measure of non-triangulability of Kn − G for a given G.