Dominating sequences in grid-like and toroidal graphs

A longest sequence $S$ of distinct vertices of a graph $G$ such that each vertex of $S$ dominates some vertex that is not dominated by its preceding vertices, is called a Grundy dominating sequence; the length of $S$ is the Grundy domination number of $G$. In this paper we study the Grundy domination number in the four standard graph products: the Cartesian, the lexicographic, the direct, and the strong product. For each of the products we present a lower bound for the Grundy domination number which turns out to be exact for the lexicographic product and is conjectured to be exact for the strong product. In most of the cases exact Grundy domination numbers are determined for products of paths and/or cycles.Comment: 17 pages 3 figure

GRUNDY DOMINATION SEQUENCES IN GENERALIZED CORONA PRODUCTS OF GRAPHS

For a graph $G=(V,E)$, a sequence $S=(v_1, v_2, \cdots, v_k)$ of distinct vertices of $G$ is called \emph{dominating sequence} if $N_G[v_i]\setminus \bigcup_{j=1}^{i-1}N[v_j]\neq\varnothing$ and is called \emph{total dominating sequence} if $N_G(v_i)\setminus \bigcup_{j=1}^{i-1}N(v_j)\neq\varnothing$ for each $2\leq i\leq k$. The maximum length of (total) dominating sequence is denoted by ($\gamma_{gr}^t)\gamma_{gr}(G)$. In this paper we compute (total) dominating sequence numbers for generalized corona products of graphs

Total dominating sequences in trees, split graphs, and under modular decomposition

A sequence of vertices in a graph G with no isolated vertices is called a total dominating sequence if every vertex in the sequence totally dominates at least one vertex that was not totally dominated by preceding vertices in the sequence, and, at the end all vertices of G are totally dominated (by definition a vertex totally dominates its neighbors). The maximum length of a total dominating sequence is called the Grundy total domination number, γgr t(G), of G, as introduced in Brešar et al. (2016). In this paper we continue the investigation of this concept, mainly from the algorithmic point of view. While it was known that the decision version of the problem is NP-complete in bipartite graphs, we show that this is also true if we restrict to split graphs. A linear time algorithm for determining the Grundy total domination number of an arbitrary forest T is presented, based on the formula γgr t(T)=2τ(T), where τ(T) is the vertex cover number of T. A similar efficient algorithm is presented for bipartite distance-hereditary graphs. Using the modular decomposition of a graph, we present a frame for obtaining polynomial algorithms for this problem in classes of graphs having relatively simple modular subgraphs. In particular, a linear algorithm for determining the Grundy total domination number of P4-tidy graphs is presented. In addition, we prove a realization result by exhibiting a family of graphs Gk such that γgr t(Gk)=k, for any k∈Z+∖{1,3}, and showing that there are no graphs G with γgr t(G)∈{1,3}. We also present such a family, which has minimum possible order and size among all graphs with Grundy total domination number equal to k.Fil: Brešar, Boštjan. University of Maribor; Eslovenia. Institute of Mathematics, Physics and Mechanics; EsloveniaFil: Kos, Tim. Institute of Mathematics, Physics and Mechanics; EsloveniaFil: Nasini, Graciela Leonor. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Torres, Pablo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentin