Distance kk-domination, distance kk-guarding, and distance kk-vertex cover of maximal outerplanar graphs

For positive integers kk and nn, let γk(n)γk(n), gk(n)gk(n), and βk(n)βk(n) denote the maximum values of the distance kk-domination number, the distance kk-guarding number, and the distance kk-vertex cover number of maximal outerplanar graphs of order nn, respectively. Known results imply γ1(n)=⌊n/3⌋γ1(n)=⌊n/3⌋ for n≥3n≥3 (Matheson et al., 1996), g1(n)=⌊n/3⌋g1(n)=⌊n/3⌋ for n≥3n≥3 (Chvátal, 1975), γ2(n)=g2(n)=⌊n/5⌋γ2(n)=g2(n)=⌊n/5⌋ for n≥5n≥5, and β2(n)=⌊n/4⌋β2(n)=⌊n/4⌋ for n≥4n≥4 (Canales et al., 2015).We show γk(n)=gk(n)=⌊n/(2k+1)⌋γk(n)=gk(n)=⌊n/(2k+1)⌋ for n≥2k+1n≥2k+1, and βk(n)=⌊n/2k⌋βk(n)=⌊n/2k⌋ for n≥2k≥4n≥2k≥4.