Bounds for the connected domination number of maximal outerplanar graphs

A dominating set of a graph G is a set S subset of 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 gamma(c)(G). Zhuang showed that gamma(c)(G) of a maximal outerplanar graph G is bounded by min{left perpendicular n+k/2 right perpendicular - 2, left perpendicular 2(n-k)/3 right perpendicular} (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 gamma(c)(G) <= left perpendicular n-k+x/2 right perpendicular, where x is a counter in the algorithm and x <= k - 2. As a corollary, the result that gamma(c)(G) <= left perpendicular n-2/2 right perpendicular for a maximal outerplanar G is gotten directly. This results is better than the above known bound for 3 < k < n+6/4. In addition, we complement some analysis with simulations to evaluate the advantages of our results. (C) 2022 Elsevier B.V. All rights reserved.