On triangle cover contact graphs

Let S={p1,p2,…,pn} be a set of pairwise disjoint geometric objects of some type in a 2D plane and let C={c1,c2,…,cn} be a set of closed objects of some type in the same plane with the property that each element in C covers exactly one element in S and any two elements in C are interior-disjoint. We call an element in S a seed and an element in C a cover. A cover contact graph (CCG) has a vertex for each element of C and an edge between two vertices whenever the corresponding cover elements touch. It is known how to construct, for any given point seed set, a disk or triangle cover whose contact graph is 1- or 2-connected but the problem of deciding whether a k-connected CCG can be constructed or not for k>2 is still unsolved. A triangle cover contact graph (TCCG) is a cover contact graph whose cover elements are triangles. In this paper, we give algorithms to construct a 3-connected TCCG and a 4-connected TCCG for a given set of point seeds. We also show that any connected outerplanar graph has a realization as a TCCG on a given set of collinear point seeds. Note that, under this restriction, only trees and cycles are known to be realizable as CCG.