[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

Common extremal graphs for three inequalities involving domination parameters

‎Let $delta (G)$‎, ‎$Delta (G)$ and $gamma(G)$‎ ‎be the minimum degree‎, ‎maximum degree and‎ ‎domination number of a graph $G=(V(G)‎, ‎E(G))$‎, ‎respectively‎. ‎A partition of $V(G)$‎, ‎all of whose classes are dominating sets in $G$‎, ‎is called a domatic partition of $G$‎. ‎The maximum number of classes of‎ ‎a domatic partition of $G$ is called the domatic number of $G$‎, ‎denoted $d(G)$‎. ‎It is well known that‎ ‎$d(G) leq delta(G)‎ + ‎1$‎, ‎$d(G)gamma(G) leq |V(G)|$ cite{ch}‎, ‎and $|V(G)| leq (Delta(G)‎+‎1)gamma(G)$ cite{berge}‎. ‎In this paper‎, ‎we investigate the graphs $G$ for which‎ ‎all the above inequalities become simultaneously equalities‎

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

Vertex domination of generalized Petersen graphs

AbstractIn a graph G a vertex v dominates all its neighbors and itself. A set D of vertices of G is (vertex) dominating set if each vertex of G is dominated by at least one vertex in D. The (vertex) domination number of G, denoted by γ(G), is the cardinality of a minimum dominating set of G. A set D of vertices in G is efficient dominating set if every vertex of G is dominated by exactly one vertex of D. For natural numbers n and k, where n>2k, a generalized Petersen graphP(n,k) is obtained by letting its vertex set be {u1,u2,…,un}∪{v1,v2,…,vn} and its edge set be the union of {uiui+1,uivi,vivi+l} over 1≤i≤n, where subscripts are reduced modulo n. We prove a necessary and sufficient condition for these graphs to have an efficient dominating set, and we determine exact values of γ(P(n,k)) for k∈{1,2,3}. Also we prove that for an odd number k, γ(P(n,k))=n2+O(k) and for an even number k>2, γ(P(n,k))≤5n9+O(k)