Due to their importance in practice, dominating set problems in graphs have been greatly studied in past and different formulations of these problems are presented in literature. This paper's focus is on two problems: weakly convex dominating set problem (WCVXDSP) and convex dominating set problem (CVXDSP). It introduces two integer linear programming (ILP) formulation for CVXDSP and one ILP mode for WCVXDSP, as well as proof for equivalency between ILP models for CVXDSP. The proof of correctness for all introduced ILP formulations is provided by showing that optimal solution to the each ILP formulation is equal to the optimal solution of the original problem.
A set @math is weakly convex in @math if for any two vertices @math there exists an @math --geodesic such that all of its vertices belong to @math . A set @math is a weakly convex dominating set if @math is weakly convex and dominating. The weakly convex domination number @math of a graph @math equals the minimum cardinality of a weakly convex dominating set in @math . The weakly convex domination subdivision number sd @math is the minimum number of edges that must be subdivided (each edge in @math can be subdivided at most once) in order to increase the weakly convex domination number. In this paper we initiate the study of weakly convex domination subdivision number and we establish upper bounds for it. We study the influence of edge subdivision on the convex domination number. We show that in general an edge subdivision can arbitrarily increase and arbitrarily decrease the convex domination number. We also find some bounds for unicyclic graphs and we investigate graphs G for which the convex domination number changes after subdivision of any edge in G.