Realizability of Graphs as Triangle Cover Contact Graphs

Let \(S=\{p_1,p_2,\ldots ,p_n\}\) be a set of pairwise disjoint geometric objects of some type and let \(C=\{c_1,c_2,\ldots ,c_n\}\) 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 \(a\times b\) grid graphs have realizations as TCCGs on collinear seeds. We also show that every complete graph has a realization as a TCCG 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.