On triangle cover contact graphs

Let S = {P-1, p(2), ..., p(n)} be a set of pairwise disjoint geometric objects of some type in a 2D plane and let C = {c(1), c(2), ..., c(n)} be a set of dosed 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. (C) 2017 Elsevier B.V. All rights reserved.ICT Fellowship, Ministry of Posts, Telecommunications and IT, Government of the People's Republic of Bangladesh [56.00.0000.028.33.007.14-254]12 month embargo; published online: 20 November 2017This item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]