Untangling circular drawings: Algorithms and complexity

Sujoy Bhore, Guangping Li, Martin Nöllenburg, Ignaz Rutter, Hsiang-Yun Wu

Abstract: We consider the problem of untangling a given (non-planar) straight-line circular drawing δG<math><msub is="true"><mrow is="true"><mi is="true">δ</mi></mrow><mrow is="true"><mi is="true">G</mi></mrow></msub></math> of an outerplanar 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> into a planar straight-line circular drawing of G by shifting a minimum number of vertices to a new position on the circle. For an outerplanar graph G, it is obvious that such a crossing-free circular drawing always exists and we define the circular shifting number shift∘(δG)<math><msup is="true"><mrow is="true"><mi mathvariant="normal" is="true">shift</mi></mrow><mrow is="true"><mo is="true">∘</mo></mrow></msup><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">δ</mi></mrow><mrow is="true"><mi is="true">G</mi></mrow></msub><mo stretchy="false" is="true">)</mo></math> as the minimum number of vertices that are required to be shifted in order to resolve all crossings of δG<math><msub is="true"><mrow is="true"><mi is="true">δ</mi></mrow><mrow is="true"><mi is="true">G</mi></mrow></msub></math>. We show that the problem Circular Untangling, asking whether shift∘(δG)≤K<math><msup is="true"><mrow is="true"><mi mathvariant="normal" is="true">shift</mi></mrow><mrow is="true"><mo is="true">∘</mo></mrow></msup><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">δ</mi></mrow><mrow is="true"><mi is="true">G</mi></mrow></msub><mo stretchy="false" is="true">)</mo><mo is="true">≤</mo><mi is="true">K</mi></math> for a given integer K, is NP-complete. For n-vertex outerplanar graphs, we obtain a tight upper bound of shift∘(δG)≤n−⌊n−2⌋−2<math><msup is="true"><mrow is="true"><mi mathvariant="normal" is="true">shift</mi></mrow><mrow is="true"><mo is="true">∘</mo></mrow></msup><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">δ</mi></mrow><mrow is="true"><mi is="true">G</mi></mrow></msub><mo stretchy="false" is="true">)</mo><mo is="true">≤</mo><mi is="true">n</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mo stretchy="false" is="true">⌊</mo><msqrt is="true"><mrow is="true"><mi is="true">n</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">2</mn></mrow></msqrt><mo stretchy="false" is="true">⌋</mo><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">2</mn></math>. Moreover, we study the Circular Untangling for almost-planar circular drawings, in which a single edge is involved in all of the crossings. For this problem, we provide a tight upper bound shift∘(δG)≤⌊n2⌋−1<math><msup is="true"><mrow is="true"><mi mathvariant="normal" is="true">shift</mi></mrow><mrow is="true"><mo is="true">∘</mo></mrow></msup><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">δ</mi></mrow><mrow is="true"><mi is="true">G</mi></mrow></msub><mo stretchy="false" is="true">)</mo><mo is="true">≤</mo><mo stretchy="false" is="true">⌊</mo><mfrac is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac><mo stretchy="false" is="true">⌋</mo><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn></math> and present an O(n2)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math>-time algorithm to compute the circular shifting number of almost-planar drawings.