Bounds for the connected domination number of maximal outerplanar graphs - Chen, Shao-Liang; Hao, Rong-Xia; Qin, Xiao-Wen (Discrete Applied Mathematics, 2022)
A dominating set of a graph G is a set S⊆V(G) such that every vertex in G is either in S or adjacent to a vertex in S. A dominating set S is a connected dominating set if the subgraph of G induced by S is connected. The minimum cardinality of a connected dominating set of G is the connected domination number, denoted by γc(G). Zhuang showed that γc(G) of a maximal outerplanar graph G is bounded by min{⌊n+k2⌋−2,⌊2(n−k)3⌋} (Zhuang, 2020), where k is the number of 2-degree vertices in G. In this paper, we give an algorithm for finding a connected dominating set of maximal outerplanar graphs and get an upper bound γc(G)≤⌊n−k+x2⌋, where x is a counter in the algorithm and x≤k−2. As a corollary, the result that γc(G)≤⌊n−22⌋ for a maximal outerplanar G is gotten directly. This results is better than the above known bound for 3<k<n+64. In addition, we complement some analysis with simulations to evaluate the advantages of our results.