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Generating Internally Triconnected Rooted Graphs

A biconnected plane graph G is called internally triconnected if any cut-pair consists of outer vertices and its removal results in only components each of which contains at least one outer vertex. In a rooted plane graph, an edge is designated as an outer edge with a specified direction. For given positive integers n ≥ 1 and g ≥ 3, let G3(n, g) (resp., Gint(n, g)) denote the class of all triconnected (resp., internally triconnected) rooted plane graphs with exactly n vertices such that the size of each inner face is at most g. In this paper, we present an efficient algorithm that enumerates all rooted plane graphs in Gint(n, g)− G3(n, g) in O(n) space and in O(1)-time delay.