Four-regular graphs with extremal rigidity properties

Tibor Jordán, Robin Huang, Henry Simmons, Kaylee Weatherspoon, Zeyu Zheng

Abstract: A graph G=(V,E)<math><mi is="true">G</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mo stretchy="false" is="true">(</mo><mi is="true">V</mi><mo is="true">,</mo><mi is="true">E</mi><mo stretchy="false" is="true">)</mo></math> is called k-edge rigid (k-edge globally rigid, resp.), if it stays rigid (globally rigid, resp.) after the deletion of at most k−1<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn></math> edges. We can define k-vertex rigidity and k-vertex global rigidity in a similar manner. It is known that if G is 3-edge rigid (2-edge globally rigid, 2-vertex globally rigid) with |V|≥5<math><mo stretchy="false" is="true">|</mo><mi is="true">V</mi><mo stretchy="false" is="true">|</mo><mo is="true">≥</mo><mn is="true">5</mn></math> then |E|≥2|V|<math><mo stretchy="false" is="true">|</mo><mi is="true">E</mi><mo stretchy="false" is="true">|</mo><mo is="true">≥</mo><mn is="true">2</mn><mo stretchy="false" is="true">|</mo><mi is="true">V</mi><mo stretchy="false" is="true">|</mo></math> holds. Furthermore, the graphs that satisfy the edge count with equality are all 4-regular.In this paper we show that for a 4-regular graph G the properties of 3-edge rigidity, 2-edge global rigidity, and essential 6-edge connectivity are equivalent. By sharpening a result of H. Fleischner, F. Genest, and B. Jackson we give a new inductive construction for the family of 4-regular and essentially 6-edge connected graphs (and hence also for the 4-regular graphs with these rigidity properties). We prove that G is 2-vertex globally rigid if and only if it is 4-vertex connected and essentially 6-edge connected.We also consider 2-vertex rigid graphs G=(V,E)<math><mi is="true">G</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mo stretchy="false" is="true">(</mo><mi is="true">V</mi><mo is="true">,</mo><mi is="true">E</mi><mo stretchy="false" is="true">)</mo></math> with minimum size |E|=2|V|−1<math><mo stretchy="false" is="true">|</mo><mi is="true">E</mi><mo stretchy="false" is="true">|</mo><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">2</mn><mo stretchy="false" is="true">|</mo><mi is="true">V</mi><mo stretchy="false" is="true">|</mo><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn></math> as well as with |E|=2|V|<math><mo stretchy="false" is="true">|</mo><mi is="true">E</mi><mo stretchy="false" is="true">|</mo><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">2</mn><mo stretchy="false" is="true">|</mo><mi is="true">V</mi><mo stretchy="false" is="true">|</mo></math>. In the former case we use our results on essentially 6-edge connected graphs to develop a new inductive construction, complementing an earlier, different construction of B. Servatius. In the latter case we characterize the edge pairs of G whose deletion preserves rigidity, and use this result to verify the correctness of a construction of 3-vertex rigid graphs on |V|≥6<math><mo stretchy="false" is="true">|</mo><mi is="true">V</mi><mo stretchy="false" is="true">|</mo><mo is="true">≥</mo><mn is="true">6</mn></math> vertices and with |E|=2|V|+2<math><mo stretchy="false" is="true">|</mo><mi is="true">E</mi><mo stretchy="false" is="true">|</mo><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">2</mn><mo stretchy="false" is="true">|</mo><mi is="true">V</mi><mo stretchy="false" is="true">|</mo><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mn is="true">2</mn></math> edges, proposed by S.A. Motevallian, C. Yu, and B.D.O. Anderson.