A note on domination number in maximal outerplanar graphs

Abstract

A set D⊆V(G) of a graph G is a dominating set if every vertex in V(G)∖D has a neighbor in D. The minimum cardinality of a dominating set of G is the domination number of G, denoted by γ(G). Let G be a maximal outerplanar graph of order n. The authors, in the article “On dominating sets of maximal outerplanar and planar graphs. Discrete Applied Mathematics, 198(2016):164-169”, proved that γ(G)≤⌊k+n4⌋ when k>0, where k is the number of bad vertices. However, we found that this result is incomplete. In this note, we construct a class An of order n maximal outerplanar graphs such that k=1 and γ(G)>⌊k+n4⌋. In addition, we show that the upper bound ⌊k+n4⌋ holds for every order n maximal outerplanar graph G satisfying k>0 and G∉An, which improves the previous conclusion.