Partitioning the vertex set of GG to make G □ HG\,\Box\, H an efficient open domination graph

A graph is an efficient open domination graph if there exists a subset of vertices whose open neighborhoods partition its vertex set. We characterize those graphs $G$ for which the Cartesian product $G \Box H$ is an efficient open domination graph when $H$ is a complete graph of order at least 3 or a complete bipartite graph. The characterization is based on the existence of a certain type of weak partition of $V(G)$. For the class of trees when $H$ is complete of order at least 3, the characterization is constructive. In addition, a special type of efficient open domination graph is characterized among Cartesian products $G \Box H$ when $H$ is a 5-cycle or a 4-cycle.Comment: 16 pages, 2 figure

Some Results on [1, k]-sets of Lexicographic Products of Graphs

A subset $S \subseteq V$ in a graph $G = (V,E)$ is called a $[1, k]$-set, if for every vertex $v \in V \setminus S$, $1 \leq | N_G(v) \cap S | \leq k$. The $[1,k]$-domination number of $G$, denoted by $\gamma_{[1, k]}(G)$ is the size of the smallest $[1,k]$-sets of $G$. A set $S'\subseteq V(G)$ is called a total $[1,k]$-set, if for every vertex $v \in V$, $1 \leq | N_G(v) \cap S | \leq k$. If a graph $G$ has at least one total $[1, k]$-set then the cardinality of the smallest such set is denoted by $\gamma_{t[1, k]}(G)$. We consider $[1, k]$-sets that are also independent. Note that not every graph has an independent $[1, k]$-set. For graphs having an independent $[1, k]$-set, we define $[1, k]$-independence numbers which is denoted by $\gamma_{i[1, k]}(G)$. In this paper, we investigate the existence of $[1,k]$-sets in lexicographic products $G\circ H$. Furthermore, we completely characterize graphs which their lexicographic product has at least one total $[1,k]$-set. Also, we determine $\gamma_{[1, k]}(G\circ H)$, $\gamma_{t[1, k]}(G\circ H)$ and $\gamma_{i[1, k]}(G\circ H)$. Finally, we show that finding smallest total $[1, k]$-set is $NP$-complete

Broadcast Domination of Triangular Matchstick Graphs and the Triangular Lattice

Blessing, Insko, Johnson and Mauretour gave a generalization of the domination number of a graph $G=(V,E)$ called the $(t,r)$ broadcast domination number which depends on the positive integer parameters $t$ and $r$. In this setting, a vertex $v \in V$ is a broadcast vertex of transmission strength $t$ if it transmits a signal of strength $t-d(u,v)$ to every vertex $u \in V$, where $d(u,v)$ denotes the distance between vertices $u$ and $v$ and $d(u,v) <t$. Given a set of broadcast vertices $S\subseteq V$, the reception at vertex $u$ is the sum of the transmissions from the broadcast vertices in $S$. The set $S \subseteq V$ is called a $(t,r)$ broadcast dominating set if every vertex $u \in V$ has a reception strength $r(u) \geq r$ and for a finite graph $G$ the cardinality of a smallest broadcast dominating set is called the $(t,r)$ broadcast domination number of $G$. In this paper, we consider the infinite triangular grid graph and define efficient $(t,r)$ broadcast dominating sets as those broadcasts that minimize signal waste. Our main result constructs efficient $(t,r)$ broadcasts on the infinite triangular lattice for all $t\geq r\geq 1$. Using these broadcasts, we then provide upper bounds for the $(t,r)$ broadcast domination numbers for triangular matchstick graphs when $(t,r)\in\{(2,1),(3,1),(3,2),(4,1),(4,2),(4,3),(t,t)\}$.Comment: 18 pages, 19 figures, 1 tabl

Domination Polynomials of Graph Products

The domination polynomials of binary graph operations, aside from union, join and corona, have not been widely studied. We compute and prove recurrence formulae and properties of the domination polynomials of families of graphs obtained by various products, ranging from explicit formulae and recurrences for specific families to more general results. As an application, we show the domination polynomial is computationally hard to evaluate.Comment: 17 page

Roman domination in Cartesian product graphs and strong product graphs

A set $S$ of vertices of a graph $G$ is a dominating set for $G$ if every vertex outside of $S$ is adjacent to at least one vertex belonging to $S$. The minimum cardinality of a dominating set for $G$ is called the domination number of $G$. A map $f : V \rightarrow \{0, 1, 2\}$ is a Roman dominating function on a graph $G$ if for every vertex $v$ with $f(v) = 0$, there exists a vertex $u$, adjacent to $v$, such that $f(u) = 2$. The weight of a Roman dominating function is given by $f(V) =\sum_{u\in V}f(u)$. The minimum weight of a Roman dominating function on $G$ is called the Roman domination number of $G$. In this article we study the Roman domination number of Cartesian product graphs and strong product graphs. More precisely, we study the relationships between the Roman domination number of product graphs and the (Roman) domination number of the factors

Cops and Robber Game with a Fast Robber on Expander Graphs and Random Graphs

We consider a variant of the Cops and Robber game, in which the robber has unbounded speed, i.e. can take any path from her vertex in her turn, but she is not allowed to pass through a vertex occupied by a cop. Let c_{infty}(G) denote the number of cops needed to capture the robber in a graph G in this variant. We characterize graphs G with c_{infty}(G)=1, and give an O(|V(G)|^2) algorithm for their detection. We prove a lower bound for c_{infty} of expander graphs, and use it to prove three things. The first is that if np > 4.2 log n then the random graph G = G(n,p) asymptotically almost surely has e1/p < c_{infty}(G) < e2 log (np)/p, for suitable constants e1 and e2. The second is that a fixed-degree random regular graph G with n vertices asymptotically almost surely has c_{infty}(G) = Theta(n). The third is that if G is a Cartesian product of m paths, then n / 4km^2 < c_{infty}(G) < n / k, where n=|V(G)| and k is the number of vertices of the longest path.Comment: 22 page

Domination Cover Number of Graphs

A set $D \subseteq V$ for the graph $G=(V, E)$ is called a dominating set if any vertex $v\in V\setminus D$ has at least one neighbor in $D$. Fomin et al.[9] gave an algorithm for enumerating all minimal dominating sets with $n$ vertices in $O(1.7159^n)$ time. It is known that the number of minimal dominating sets for interval graphs and trees on $n$ vertices is at most $3^{n/3} \approx 1.4422^n$. In this paper, we introduce the domination cover number as a new criterion for evaluating the dominating sets in graphs. The domination cover number of a dominating set $D$, denoted by $\mathcal{C}_D(G)$, is the summation of the degrees of the vertices in $D$. Maximizing or minimizing this parameter among all minimal dominating sets have interesting applications in many real-world problems, such as the art gallery problem. Moreover, we investigate this concept for different graph classes and propose some algorithms for finding the domination cover number in trees, block graphs

A general lower bound for the domination number of cylindrical graphs

In this paper we present a lower bound for the domination number of the Cartesian product of a path and a cycle, that is tight if the length of the cycle is a multiple of five. This bound improves the natural lower bound obtained by using the domination number of the Cartesian product of two paths, that is the best one known so far.Comment: 12 pages, 3 figure

Quasi-efficient domination in grids

Domination of grids has been proved to be a demanding task and with the addition of independence it becomes more challenging. It is known that no grid with $m,n \geq 5$ has an efficient dominating set, also called perfect code, that is, an independent vertex set such that each vertex not in it has exactly one neighbor in that set. So it is interesting to study the existence of independent dominating sets for grids that allow at most two neighbors, such sets are called independent $[1,2]$-sets. In this paper we prove that every grid has an independent $[1,2]$-set, and we develop a dynamic programming algorithm using min-plus algebra that computes $i_{[1,2]}(P_m\Box P_n)$, the minimum cardinality of an independent $[1,2]$-set for the grid graph $P_m\square P_n$. We calculate $i_{[1,2]}(P_m\Box P_n)$ for $2\leq m\leq 13, n\geq m$ using this algorithm, meanwhile the parameter for grids with $14\leq m\leq n$ is obtained through a quasi-regular pattern that, in addition, provides an independent $[1,2]$-set of minimum size.Comment: 17 pages, 16 figure

Algorithmic problems in right-angled Artin groups: complexity and applications

In this paper we consider several classical and novel algorithmic problems for right-angled Artin groups, some of which are closely related to graph theoretic problems, and study their computational complexity. We study these problems with a view towards applications to cryptography.Comment: 16 page