The Domination Complexity and Related Extremal Values of Large 3D Torus

Abstract

Domination is a structural complexity of chemical molecular graphs. A dominating set in a (molecular) graph G=V,E is a subset S⊆V such that each vertex in V\S is adjacent to at least one vertex in S. The domination number γG of a graph G is the minimum size of a dominating set in G. In this paper, computer-aided approaches for obtaining bounds for domination number on torus graphs are here considered, and many new exact values and bounds are obtained.

Hamilton-connectivity of Interconnection Networks Modeled by a Product of Graphs

“…A perfect matching between two disjoint graphs G 1 , G 2 with the same order n is a matching consisting of n edges such that each of them has one end vertex in G 1 and the other one in G 2 . See [7][8][9][10][11][12][13][14].…”

More Results on the Domination Number of Cartesian Product of Two Directed Cycles

“…Graph domination and associated concepts have been studied for many years and there are more than 200 papers to study on the subject [2][3][4]. Among them, many authors study the domination number of products of graphs [5,6], especially for cylinders [7], torus [8,9] and grids [10]. Liu et al initiated the study of the domination number of two directed cycles, and they [11,12] determined the exact values of γ(C m C n ) for m up to 6 and showed that Theorem 1. γ(C m C n ) = mn 3 if m ≡ 0 (mod 3) and n ≡ 0 (mod 3).…”

The Double Roman Domination Numbers of Generalized Petersen Graphs P(n, 2)

“…There have been more than 200 papers studying various domination on graphs in the literature [2][3][4][5][6]. Among them, Roman domination and double Roman domination appear to be a new variety of interest [3,[7][8][9][10][11][12][13][14][15].…”