Some inequalities about connected domination number

AbstractLet G = (V,E) be a graph. In this note, γc, ir, γ, i, β0, Γ, IR denote the connected domination number, the irredundance number, the domination number, the independent domination number, the independence number, the upper domination number and the upper irredundance number, respectively. We prove that γc ⩽ 3 ir − 2 for a connected graph G. Thus, an open problem in Hedetniemi and Laskar (1984) discuss further some relations between γc and γ, β0, Γ, IR, respectively

Connected domination in graphs and v-numbers of binomial edge ideals

The v-number of a graded ideal is an algebraic invariant introduced by Cooper et al., and originally motivated by problems in algebraic coding theory. In this paper we study the case of binomial edge ideals and we establish a significant connection between their v-numbers and the concept of connected domination in graphs. More specifically, we prove that the localization of the v-number at one of the minimal primes of the binomial edge ideal $J_G$ of a graph $G$ coincides with the connected domination number of the defining graph, providing a first algebraic description of the connected domination number. As an immediate corollary, we obtain a sharp combinatorial upper bound for the v-number of binomial edge ideals of graphs. Lastly, building on some known results on edge ideals, we analyze how the v-number of $J_G$ behaves under Gr\"obner degeneration when $G$ is a closed graph.Comment: 19 pages. Comments welcom