The exact domination number of the generalized Petersen graphs

AbstractLet G=(V,E) be a graph. A subset S⊆V is a dominating set of G, if every vertex u∈V−S is dominated by some vertex v∈S. The domination number, denoted by γ(G), is the minimum cardinality of a dominating set. For the generalized Petersen graph G(n), Behzad et al. [A. Behzad, M. Behzad, C.E. Praeger, On the domination number of the generalized Petersen graphs, Discrete Mathematics 308 (2008) 603–610] proved that γ(G(n))≤⌈3n5⌉ and conjectured that the upper bound ⌈3n5⌉ is the exact domination number. In this paper we prove this conjecture

[1,2]-Domination in Generalized Petersen Graphs

A vertex subset $S$ of a graph $G=(V,E)$ is a $[1,2]$-dominating set if each vertex of $V\backslash S$ is adjacent to either one or two vertices in $S$. The minimum cardinality of a $[1,2]$-dominating set of $G$, denoted by $\gamma_{[1,2]}(G)$, is called the $[1,2]$-domination number of $G$. In this paper the $[1,2]$-domination and the $[1,2]$-total domination numbers of the generalized Petersen graphs $P(n,2)$ are determined

Advances in Discrete Applied Mathematics and Graph Theory

The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs