From the quasi-total strong differential to quasi-total italian domination in graphs

Identifier:  imarina:9219146

Handle: https://hdl.handle.net/20.500.11797/imarina9219146

Authors:  Cabrera Martinez, Abel; Estrada-Moreno, Alejandro; Rodriguez-Velazquez, Juan Alberto

Abstract:
This paper is devoted to the study of the quasi-total strong differential of a graph, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. Given a vertex x ∈ V(G) of a graph G, the neighbourhood of x is denoted by N(x). The neighbourhood of a set X ⊆ V(G) is defined to be N(X) =⋃ x∈X N(x), while the external neighbourhood of X is defined to be Ne (X) = N(X) \ X. Now, for every set X ⊆ V(G) and every vertex x ∈ X, the external private neighbourhood of x with respect to X is defined as the set Pe (x, X) = {y ∈ V(G) \ X: N(y) ∩ X = {x}}. Let Xw = {x ∈ X: Pe (x, X) ̸= ∅}. The strong differential of X is defined to be ∂s (X) = |Ne (X)| − |Xw |, while the quasi-total strong differential of G is defined to be ∂s ∗ (G) = max{∂s (X): X ⊆ V(G) and Xw ⊆ N(X)}. We show that the quasi-total strong differential is closely related to several graph parameters, including the domination number, the total domination number, the 2-domination number, the vertex cover number, the semitotal domination number, the strong differential, and the quasi-total Italian domination number. As a consequence of the study, we show that the problem of finding the quasi-total strong differential of a graph is NP-hard.