Mutual and total mutual visibility in hypercube-like graphs

Serafino Cicerone, Alessia Di Fonso, Gabriele Di Stefano, Alfredo Navarra, Francesco Piselli

Abstract: Highlights•We study the mutual-visibility and the total mutual-visibility in hypercube-like graphs.•For a hypercube Qd<math><msub is="true"><mrow is="true"><mi is="true">Q</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub></math>, we show that (d⌊d2⌋)+(d⌊d2⌋+3)≤μ(Qd)≤2d−1<math><mrow is="true"><mo is="true">(</mo><mtable is="true"><mtr is="true"><mtd is="true"><mi is="true">d</mi></mtd></mtr><mtr is="true"><mtd is="true"><mrow is="true"><mo stretchy="false" is="true">⌊</mo><mfrac is="true"><mrow is="true"><mi is="true">d</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac><mo stretchy="false" is="true">⌋</mo></mrow></mtd></mtr></mtable><mo is="true">)</mo></mrow><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mrow is="true"><mo is="true">(</mo><mtable is="true"><mtr is="true"><mtd is="true"><mi is="true">d</mi></mtd></mtr><mtr is="true"><mtd is="true"><mrow is="true"><mo stretchy="false" is="true">⌊</mo><mfrac is="true"><mrow is="true"><mi is="true">d</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac><mo stretchy="false" is="true">⌋</mo><mo is="true">+</mo><mn is="true">3</mn></mrow></mtd></mtr></mtable><mo is="true">)</mo></mrow><mo is="true">≤</mo><mi is="true">μ</mi><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">Q</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub><mo stretchy="false" is="true">)</mo><mo is="true">≤</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">d</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">1</mn></mrow></msup></math>, and an O(d)<math><mi is="true">O</mi><mo stretchy="false" is="true">(</mo><msqrt is="true"><mrow is="true"><mi is="true">d</mi></mrow></msqrt><mo stretchy="false" is="true">)</mo></math>-approximation algorithm for computing μ(Qd)<math><mi is="true">μ</mi><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">Q</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub><mo stretchy="false" is="true">)</mo></math> is given.•For a Fibonacci cube Γd<math><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">Γ</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub></math>, we provide optimal solutions for μ(Γd)<math><mi is="true">μ</mi><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">Γ</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub><mo stretchy="false" is="true">)</mo></math> when d≤6<math><mi is="true">d</mi><mo is="true">≤</mo><mn is="true">6</mn></math>, whereas for larger values of d, we prove μ(Γd)≤Fd+Fd−2<math><mi is="true">μ</mi><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">Γ</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub><mo stretchy="false" is="true">)</mo><mo is="true">≤</mo><msub is="true"><mrow is="true"><mi is="true">F</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><msub is="true"><mrow is="true"><mi is="true">F</mi></mrow><mrow is="true"><mi is="true">d</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">2</mn></mrow></msub></math> and μ(Γd)≥max0≤k≤⌈d2⌉⁡(d−k+1k)<math><mi is="true">μ</mi><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">Γ</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub><mo stretchy="false" is="true">)</mo><mo is="true">≥</mo><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">max</mi></mrow><mrow is="true"><mn is="true">0</mn><mo is="true">≤</mo><mi is="true">k</mi><mo is="true">≤</mo><mo stretchy="false" is="true">⌈</mo><mfrac is="true"><mrow is="true"><mi is="true">d</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac><mo stretchy="false" is="true">⌉</mo></mrow></msub><mo is="true">⁡</mo><mrow is="true"><mo is="true">(</mo><mtable is="true"><mtr is="true"><mtd is="true"><mrow is="true"><mi is="true">d</mi><mo is="true">−</mo><mi is="true">k</mi><mo is="true">+</mo><mn is="true">1</mn></mrow></mtd></mtr><mtr is="true"><mtd is="true"><mi is="true">k</mi></mtd></mtr></mtable><mo is="true">)</mo></mrow></math>. Furthermore, a 1ϕ(⌈d2⌉+1)<math><mfrac is="true"><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mi is="true">ϕ</mi></mrow></mfrac><mrow is="true"><mo stretchy="true" is="true">(</mo><mo stretchy="false" is="true">⌈</mo><mfrac is="true"><mrow is="true"><mi is="true">d</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac><mo stretchy="false" is="true">⌉</mo><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mn is="true">1</mn><mo stretchy="true" is="true">)</mo></mrow></math>-approximation algorithm for computing μ(Γd)<math><mi is="true">μ</mi><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi mathvariant="normal" is="true">Γ</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub><mo stretchy="false" is="true">)</mo></math> is given.•For a cube-connected cycle CCCd<math><msub is="true"><mrow is="true"><mi mathvariant="italic" is="true">CCC</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub></math>, we show that 2⌈d2⌉≤μ(CCCd)≤3⋅2d−2<math><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mo stretchy="false" is="true">⌈</mo><mfrac is="true"><mrow is="true"><mi is="true">d</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac><mo stretchy="false" is="true">⌉</mo></mrow></msup><mo is="true">≤</mo><mi is="true">μ</mi><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi mathvariant="italic" is="true">CCC</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub><mo stretchy="false" is="true">)</mo><mo is="true">≤</mo><mn is="true">3</mn><mo is="true">⋅</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">d</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">2</mn></mrow></msup></math>. For the total mutual-visibility, μt(CCCd)=0<math><msub is="true"><mrow is="true"><mi is="true">μ</mi></mrow><mrow is="true"><mi is="true">t</mi></mrow></msub><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi mathvariant="italic" is="true">CCC</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub><mo stretchy="false" is="true">)</mo><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">0</mn></math>. Furthermore, a 3⋅2⌊d2⌋−2<math><mn is="true">3</mn><mo is="true">⋅</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mo stretchy="false" is="true">⌊</mo><mfrac is="true"><mrow is="true"><mi is="true">d</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac><mo stretchy="false" is="true">⌋</mo><mo linebreak="badbreak" linebreakstyle="after" is="true">−</mo><mn is="true">2</mn></mrow></msup></math>-approximation algorithm for computing μ(CCCd)<math><mi is="true">μ</mi><mo stretchy="false" is="true">(</mo><msub is="true"><mrow is="true"><mi mathvariant="italic" is="true">CCC</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub><mo stretchy="false" is="true">)</mo></math> is given.•For a butterfly BF(d)<math><mrow is="true"><mi mathvariant="italic" is="true">BF</mi></mrow><mo stretchy="false" is="true">(</mo><mi is="true">d</mi><mo stretchy="false" is="true">)</mo></math>, we show that μ(BF(d))=2d+1−2<math><mi is="true">μ</mi><mo stretchy="false" is="true">(</mo><mrow is="true"><mi mathvariant="italic" is="true">BF</mi></mrow><mo stretchy="false" is="true">(</mo><mi is="true">d</mi><mo stretchy="false" is="true">)</mo><mo stretchy="false" is="true">)</mo><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">d</mi><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mn is="true">1</mn></mrow></msup><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">2</mn></math> and μt(BF(d))=2d<math><mi is="true">μ</mi><mi is="true">t</mi><mo stretchy="false" is="true">(</mo><mi is="true">B</mi><mi is="true">F</mi><mo stretchy="false" is="true">(</mo><mi is="true">d</mi><mo stretchy="false" is="true">)</mo><mo stretchy="false" is="true">)</mo><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">d</mi></mrow></msup></math>.