New Bounds for Three Outer-Independent Domination-Related Parameters in Cactus Graphs
Autor
Cabrera-Martínez, Abel
Rueda-Vázquez, Juan Manuel
Segarra, Jaime
Editor
MDPIFecha
2024Materia
Total outer-independent dominationDouble outer-independent domination
2-outer-independent domination
Cactus graphs
Trees
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Let G be a nontrivial connected graph. For a set D _ V(G), we define D = V(G) n D. The set D is a total outer-independent dominating set of G if jN(v) \ Dj _ 1 for every vertex v 2 V(G) and D is an independent set of G. Moreover, D is a double outer-independent dominating set of G if jN[v] \ Dj _ 2 for every vertex v 2 V(G) and D is an independent set of G. In addition, D is a 2-outer-independent dominating set of G if jN(v) \ Dj _ 2 for every vertex v 2 D and D is an independent set of G. The total, double or 2-outer-independent domination number of G, denoted by goi t (G), goi_2(G) or goi 2 (G), is the minimum cardinality among all total, double or 2-outer-independent dominating sets of G, respectively. In this paper, we first show that for any cactus graph G of order n(G) _ 4 with k(G) cycles, goi 2 (G) _ n(G)+l(G) 2 + k(G), goi t (G) _ 2n(G)l(G)+s(G) 3 + k(G) and goi_2(G) _ 2n(G)+l(G)+s(G) 3 + k(G), where l(G) and s(G) represent the number of leaves and the number of support vertices of G, respectively. These previous bounds extend three known results given for trees. In addition, we characterize the trees T with goi_2(T) = goi t (T). Moreover, we show that goi 2 (T) _ n(T)+l(T)s(T)+1 2 for any tree T with n(T) _ 3. Finally, we give a constructive characterization of the trees T that satisfy the equality above.