Realizability of graphs as triangle cover contact graphs

Let S={p1,p2,…,pn} be a set of pairwise disjoint geometric objects of some type and let C={c1,c2,…,cn} be a set of closed objects of some type with the property that each element in C covers exactly one element in S and any two elements in C can intersect only on their boundaries. We call an element in S a seed and an element in C a cover. A cover contact graph (CCG) consists of a set of vertices and a set of edges where each of the vertex corresponds to each of the covers and each edge corresponds to a connection between two covers if and only if they touch at their boundaries. A triangle cover contact graph (TCCG) is a cover contact graph whose cover elements are triangles. In this paper, we show that every Halin graph has a realization as a TCCG on a given set of collinear seeds. We introduce a new class of graphs which we call super-Halin graphs. We also show that the classes super-Halin graphs, cubic planar Hamiltonian graphs and a×b grid graphs have realizations as TCCGs on collinear seeds. We also show that some non-planar graphs have planar realizations as TCCGs. Every complete graph and clique-3 graph have realizations as TCCGs on any given set of seeds. Note that only trees and cycles are known to be realizable as CCGs and outerplanar graphs are known to be realizable as TCCGs.