On triangle cover contact graphs

Shaheena Sultana, Md. Iqbal Hossain, Md. Saidur Rahman, Nazmun Nessa Moon, Tahsina Hashem

Abstract: Let S={p1,p2,…,pn}<math><mi is="true">S</mi><mo is="true">=</mo><mo stretchy="false" is="true">{</mo><msub is="true"><mrow is="true"><mi is="true">p</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msub><mo is="true">,</mo><msub is="true"><mrow is="true"><mi is="true">p</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><mo is="true">,</mo><mo is="true">…</mo><mo is="true">,</mo><msub is="true"><mrow is="true"><mi is="true">p</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub><mo stretchy="false" is="true">}</mo></math> be a set of pairwise disjoint geometric objects of some type in a 2D plane and let C={c1,c2,…,cn}<math><mi is="true">C</mi><mo is="true">=</mo><mo stretchy="false" is="true">{</mo><msub is="true"><mrow is="true"><mi is="true">c</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msub><mo is="true">,</mo><msub is="true"><mrow is="true"><mi is="true">c</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><mo is="true">,</mo><mo is="true">…</mo><mo is="true">,</mo><msub is="true"><mrow is="true"><mi is="true">c</mi></mrow><mrow is="true"><mi is="true">n</mi></mrow></msub><mo stretchy="false" is="true">}</mo></math> 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<math><mi is="true">k</mi><mo is="true">&gt;</mo><mn is="true">2</mn></math> is still unsolved. A triangle cover contact graph (TCCG<math><mi is="true">T</mi><mi is="true">C</mi><mi is="true">C</mi><mi is="true">G</mi></math>) is a cover contact graph whose cover elements are triangles. In this paper, we give algorithms to construct a 3-connected TCCG<math><mi is="true">T</mi><mi is="true">C</mi><mi is="true">C</mi><mi is="true">G</mi></math> and a 4-connected TCCG<math><mi is="true">T</mi><mi is="true">C</mi><mi is="true">C</mi><mi is="true">G</mi></math> for a given set of point seeds. We also show that any connected outerplanar graph has a realization as a TCCG<math><mi is="true">T</mi><mi is="true">C</mi><mi is="true">C</mi><mi is="true">G</mi></math> on a given set of collinear point seeds. Note that, under this restriction, only trees and cycles are known to be realizable as CCG.

External IDs:dblp:journals/comgeo/SultanaHRMH18