# Realizability of Graphs as Triangle Cover Contact Graphs

## Abstract

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 *TCCG*s 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 *CCG*s and outerplanar graphs are known to be realizable as *TCCG*s.

## Notes

### Acknowledgement

This work is done in Graph Drawing and Information Visualization Laboratory, Department of Computer Science and Engineering, Bangladesh University of Engineering and Technology as a part of a Ph.D. research work. The first author is supported by ICT Fellowship of ICT division, Ministry of Posts, Telecommunications and IT, Government of the People’s Republic of Bangladesh.

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