Related papers: Domination parameters and diameter of Abelian Cayl…
The study of power domination in graphs arises from the problem of placing a minimum number of measurement devices in an electrical network while monitoring the entire network. A power dominating set of a graph is a set of vertices from…
Given a directed graph $D$, a set $S \subseteq V(D)$ is a total dominating set of $D$ if each vertex in $D$ has an in-neighbor in $S$. The total domination number of $D$, denoted $\gamma_t(D)$, is the minimum cardinality among all total…
In this paper, we show that several graph parameters are known in different areas under completely different names. More specifically, our observations connect signed domination, monopolies, $\alpha$-domination, $\alpha$-independence,…
Let $s$ be a positive integer. Our goal is to find all finite abelian groups $G$ that contain a $2$-subset $A$ for which the undirected Cayley graph $\Gamma(G,A)$ has diameter at most $s$. We provide a complete answer when $G$ is cyclic,…
In this paper, we obtain the exact values of several domination parameters for the direct product of a complete graph with a path or a cycle. Specifically, we determine the domination number, independent domination number,…
We study relations between diameter $D(G)$, domination number $\gamma(G)$, independence number $\alpha(G)$ and cop number $c(G)$ of a connected graph $G$, showing (i.) $c(G) \leq \alpha(G)-\lfloor \frac{D(G)-3}{2} \rfloor$, and (ii.) $c(G)…
For an integer $k \ge 1$, a (distance) $k$-dominating set of a connected graph $G$ is a set $S$ of vertices of $G$ such that every vertex of $V(G) \setminus S$ is at distance at most~$k$ from some vertex of $S$. The $k$-domination number,…
A total dominating set of a graph G with no isolated vertices is a subset S of the vertex set such that every vertex of G is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a total dominating set of…
In this paper we study combinatorial and algorithmic resp. complexity questions of upper domination, i.e., the maximum cardinality of a minimal dominating set in a graph. We give a full classification of the related maximisation and…
This paper introduces the concept of compliant vertices and compliant graphs, with a focus on the total domination degree (TDD) of a vertex in compliant graphs. The TDD is systematically calculated for various graph classes, including path…
A power dominating set of a graph is a set of vertices that observes every vertex in the graph by combining classical domination with an iterative propagation process arising from electrical circuit theory. In this paper, we study the power…
We use probabilistic methods to find lower bounds on the maximum number, in a graph with domination number \gamma, of dominating sets of size \gamma. We find that we can randomly generate a graph that, w.h.p., is dominated by almost all…
The ratio of the connected domination number, $\gamma_c$, and the domination number, $\gamma$, is strictly bounded from above by 3. It was shown by Zverovich that for every connected $(P_5,C_5)$-free graph, $\gamma_c = \gamma$. In this…
We introduce a domination polynomial of a graph G. The domination polynomial of a graph G of order n is the polynomial D(G, x) =\sum_{i=1}^n d(G, i)x^i, where d(G, i) is the number of dominating sets of G of size i. We obtain some…
The minimum dominating set problem asks for a dominating set with minimum size. First, we determine some vertices contained in the minimum dominating set of a graph. By applying a particular scheme, we ensure that the resulting graph is…
Let $R$ be a finite commutative ring with $1\ne 0$. In this article, we study the total graph of $R$, denoted by $\tau (R)$, determine some of its basic graph-theoretical properties, determine when it is Eulerian, and find some conditions…
We propose the conjecture that the domination number $\gamma(G)$ of a $\Delta$-regular graph $G$ with $\Delta\geq 1$ is always at most its edge domination number $\gamma_e(G)$, which coincides with the domination number of its line graph.…
In this paper, we study the total outer-connected domination number of the middle graph of a simple graph and we obtain tight bounds for this number in terms of the order of the middle graph. We also compute the total outer-connected…
We study the problem of half-domination sets of vertices in vertex transitive infinite graphs generated by regular or semi-regular tessellations of the plane. In some cases, the results obtained are sharp and in the rest, we show upper…
A well-known conjecture of Babai states that if $G$ is a finite simple group and $X$ is a generating set of $G$, then the diameter of the Cayley graph $Cay(G,X)$ is bounded above by $(\log |G|)^c$ for some absolute constant $c$. The goal of…