For any graph G = (V, E) and proportion $p\in(0,1]$, a set $S\subseteq V$ is a p-dominating set if $\frac{|N[S]|}{|V|}\geq p$. The $p$-domination number $γ_{p}(G)$ equals the minimum cardinality of a $p$-dominating set in G. For a permutation $π$ of the vertex set of G, the graph $π$G is obtained from two disjoint copies $G_1$ and $G_2$ of $G$ by joining each v in $G_1$ to $π(v)$ in $G_2$. i.e., $V(πG)= V(G_1)\cup V(G_2) \text{ and } E(G)= E(G_1)\cup E(G_2)\cup \{\{v,π(v)\}: v\in V(G_1), π(v)\in V(G_2)\}$. The graph $πG$ is called the prism of $G$ with respect to $π$. In this paper, we find some relations between the domination and the $p$-domination numbers in the context of graph and its prism graph for particular values of $p$.
For a given graph $G=(V,E)$ and permutation $π:V\mapsto V$ the prism $πG$ of $G$ is defined as follows: $V(πG)=V(G)\cup V(G')$, where $G'$ is a copy of $G$, and $E(πG)=E(G)\cup E(G')\cup M_π$, where $M_π=\{uv': u\in V(G), v=π(u)\}$ and $v'$ denotes the copy of $v$ in $G'$. We study and compare the properties of convex and weakly convex dominating sets in prism graphs. In particular, we characterize prism $γ_{con}$-fixers and -doublers. We also show that the differences $γ_{wcon}(G)-γ_{wcon}(πG)$ and $γ_{wcon}(πG) - 2γ_{wcon}(G)$ can be arbitrarily large, and that the convex domination number o…
With the aid of hypergraph transversals it is proved that $γ_t(Q_{n+1}) = 2γ(Q_n)$, where $γ_t(G)$ and $γ(G)$ denote the total domination number and the domination number of $G$, respectively, and $Q_n$ is the $n$-dimensional hypercube. More generally, it is shown that if $G$ is a bipartite graph, then $γ_t(G \square K_2) = 2γ(G)$. Further, we show that the bipartite condition is essential by constructing, for any $k \ge 1$, a (non-bipartite) graph $G$ such that $γ_t (G \square K_2 ) = 2γ(G) - k$. Along the way several domination-type identities for hypercubes are also obtained.
An eternal dominating set of a graph $G$ is a set of vertices (or "guards") which dominates $G$ and which can defend any infinite series of vertex attacks, where an attack is defended by moving one guard along an edge from its current position to the attacked vertex. The size of the smallest eternal dominating set is denoted $γ^\infty(G)$ and is called the eternal domination number of $G$. In this paper, we answer a conjecture of Klostermeyer and Mynhardt [Discussiones Mathematicae Graph Theory, vol. 35, pp. 283-300], showing that there exist there are infinitely many graphs $G$ such that $γ^\…
In this paper, we define a new domination invariant on a graph $G$, which coincides with the ordinary independent domination number of the generalized prism $G \Box K_k$, called the $k$-rainbow independent domination number and denoted by $γ_{{\rm ri}k}(G)$. Some bounds and exact values concerning this domination concept are determined. As a main result, we prove a Nordhaus-Gaddum-type theorem on the sum for $2$-rainbow independent domination number, and show if G is a graph of order $n \geq 3$, then $5\leq γ_{{\rm ri}2}(G)+γ_{{\rm ri}2}(\overline{G})\leq n+3$, with both bounds being sharp.
Recently the notion of $k$-rainbow total domination was introduced for a graph $G$, motivated by a desire to reduce the problem of computing the total domination number of the generalized prism $G \Box K_k$ to an integer labeling problem on $G$. In this paper we further demonstrate usefulness of the labeling approach, presenting bounds on the rainbow total domination number in terms of the total domination number, the rainbow domination number and the rainbow total domination number, as well as the usual domination number, where the latter presents a generalization of a result by Goddard and H…
Let $G=(V,E)$ be a graph. For some $α$ with $0<α\leq 1$, a subset $S$ of $V$ is said to be a $α$-partial dominating set if $|N[S]|\geq α|V|$. The size of a smallest such $S$ is called the $α$-partial domination number and is denoted by $\mathsf{pd}_α(G)$. In this paper, we introduce $α$-partial domination number in a graph $G$ and study different bounds on the partial domination number of a graph $G$ with respect to its order, maximum degree, domination number etc., Moreover, $α$-partial domination spectrum is introduced and Nordhaus-Gaddum bounds on the partial domination number are studied.
A set $S\subseteq V$ is a dominating set of $G$ if every vertex in $V - S$ is adjacent to at least one vertex in $S$. The domination number $γ(G)$ of $G$ equals the minimum cardinality of a dominating set $S$ in $G$; we say that such a set $S$ is a $γ$-set. The single greatest focus of research in domination theory is the determination of the value of $γ(G)$. By definition, all vertices must be dominated by a $γ$-set. In this paper we propose relaxing this requirement, by seeking sets of vertices that dominate a prescribed fraction of the vertices of a graph. We focus particular attention on $…
Let $G_1$ and $G_2$ be disjoint copies of a graph $G$, and let $f: V(G_1) \rightarrow V(G_2)$ be a function. Then a \emph{functigraph} $C(G, f)=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid u \in V(G_1), v \in V(G_2), v=f(u)\}$. A functigraph is a generalization of a \emph{permutation graph} (also known as a \emph{generalized prism}) in the sense of Chartrand and Harary. In this paper, we study domination in functigraphs. Let $γ(G)$ denote the domination number of $G$. It is readily seen that $γ(G) \le γ(C(G,f)) \le 2 γ(G)$. We investig…
We prove the following result: If $G$ be a connected graph on $n \ge 6$ vertices, then there exists a set of vertices $D$ with $|D| \le \frac{n}{3}$ and such that $V(G) \setminus N[D]$ is an independent set, where $N[D]$ is the closed neighborhood of $D$. Furthermore, the bound is sharp. This seems to be the first result in the direction of partial domination with constrained structure on the graph induced by the non-dominated vertices, which we further elaborate in this paper.
Two new techniques are introduced into the theory of the domination game. The cutting lemma bounds the game domination number of a partially dominated graph with the game domination number of suitably modified partially dominated graph. The union lemma bounds the S-game domination number of a disjoint union of paths using appropriate weighting functions. Using these tools a conjecture asserting that the so-called three legged spiders are game domination critical graphs is proved. An extended cutting lemma is also derived and all game domination critical trees on 18, 19, and 20 vertices are lis…
The $k$-rainbow independent domination number of a graph $G$, denoted $γ_{\rm rik}(G)$, is the cardinality of a smallest set consisting of two vertex-disjoint independent sets $V_1$ and $V_2$ for which every vertex in $V(G)\setminus (V_1\cup V_2)$ has neighbors in both $V_1$ and $V_2$. This domination invariant was proposed by {Š}umenjak, Rall and Tepeh in (Applied Mathematics and Computation 333(15), 2018: 353-361), which allows to reduce the problem of computing the independent domination number of the generalized prism $G {\Box} K_k$ to an integer labeling problem on $G$. They proved a Nord…
For a graph $G$, a vertex subset $S \subseteq V(G)$ is said to be $K_{k}$-isolating if $G - N_{G}[S]$ does not contain $K_{k}$ as a subgraph. The $K_{k}$-isolation number of $G$, denoted by $ι_{k}(G)$, is the minimum cardinality of a $K_{k}$-isolating set of $G$. Analogously, $S$ is said to be independent $K_{k}$-isolating if $S$ is a $K_{k}$-isolating set of $G$ and $G[S]$ has no edge. The independent $K_{k}$-isolation number of $G$, denoted by $ι'_{k}(G)$, is the minimum cardinality of an independent $K_{k}$-isolating set of $G$. Clearly, when $k = 1$, we have $γ(G) = ι_{1}(G)$ and $i(G) = ι…
Several domination results have been obtained for maximal outerplanar graphs (mops). The classical domination problem is to minimize the size of a set $S$ of vertices of an $n$-vertex graph $G$ such that $G - N[S]$, the graph obtained by deleting the closed neighborhood of $S$, is null. A classical result of Chvátal is that the minimum size is at most $n/3$ if $G$ is a mop. Here we consider a modification by allowing $G - N[S]$ to have isolated vertices and isolated edges only. Let $ι_1(G)$ denote the size of a smallest set $S$ for which this is achieved. We show that if $G$ is a mop on $n \ge…
Given a graph $G$ and a subset of vertices $D\subseteq V(G)$, the external neighbourhood of $D$ is defined as $N_e(D)=\{u\in V(G)\setminus D:\, N(u)\cap D\ne \varnothing\}$, where $N(u)$ denotes the open neighbourhood of $u$. Now, given a subset $D\subseteq V(G)$ and a vertex $v\in D$, the external private neighbourhood of $v$ with respect to $D$ is defined to be $epn(v,D)=\{u\in V(G)\setminus D: N(u)\cap D=\{v\}\}.$ The strong differential of a set $D\subseteq V(G)$ is defined as $\partial_s(D)=|N_e(D)|-|D_w|,$ where $D_w=\{v\in D: epn(v,D)\neq \varnothing\}$. In this paper we focus on the st…
Let $G=(V,E)$ be a graph of order $n$ and let $γ_{R}(G)$ and $\partial (G)$ denote the Roman domination number and the differential of $G,$ respectively. In this paper we prove that for any integer $k\geq 0$, if $G$ is a graph of order $n\geq 6k+9$, minimum degree $δ\geq 2,$ which does not contain any induced $\{C_{5},C_{8},\ldots ,C_{3k+2}\}$% -cycles, then $γ_{R}(G)\leq \frac{(4k+8)n}{6k+11}$. This bound is an improvement of the bounds given in [E.W. Chambers, B. Kinnersley, N. Prince, and D.B. West, Extremal problems for Roman domination, SIAM J. Discrete Math. 23 (2009) 1575--1586] when $k…
A set $S\subseteq V$ is a dominating set of $G$ if every vertex in $V - S$ is adjacent to at least one vertex in $S$. The domination number $γ(G)$ of $G$ equals the minimum cardinality of a dominating set $S$ in $G$; we say that such a set $S$ is a $γ$-set. A generalization of this is partial domination which was introduced in 2017 by Case, Hedetniemi, Laskar, and Lipman [3,2] . In partial domination a set $S$ is a $p$-dominating set if it dominates a proportion $p$ of the vertices in $V$. The p-domination number $γ_{p}(G)$ is the minimum cardinality of a $p$-dominating set in $G$. In this pap…
Let $G$ be a graph of order $n$ and size $m$ and let $k\geq 1$ be an integer. A $k$-tuple total dominating set in $G$ is called a $k$-tuple total restrained dominating set of $G$ if each vertex $x\in V(G)-S$ is adjacent to at least $k$ vertices of $V(G)-S$. The minimum number of vertices of a such sets in $G$ are the $k$-tuple total restrained domination number $γ_{\times k,t}^{r}(G)$ of $G$. The maximum number of classes of a partition of $V(G)$ such that its all classes are $k$-tuple total restrained dominating sets in $G$, is called the $k$-tuple total restrained domatic number of $G$. In t…
Let $G$ be a graph of order $n(G)$ and vertex set $V(G)$. Given a set $S\subseteq V(G)$, we define the external neighbourhood of $S$ as the set $N_e(S)$ of all vertices in $V(G)\setminus S$ having at least one neighbour in $S$. The differential of $S$ is defined to be $\partial(S)=|N_e(S)|-|S|$. In this paper, we introduce the study of the $2$-packing differential of a graph, which we define as $\partial_{2p}(G)=\max\{\partial(S): S\subseteq V(G) \text{ is a }2\text{-packing}\}.$ We show that the $2$-packing differential is closely related to several graph parameters, including the packing num…
A dominating set of a graph $G=(V,E)$ is a vertex set $D$ such that every vertex in $V(G) \setminus D$ is adjacent to a vertex in $D$. The cardinality of a smallest dominating set of $D$ is called the domination number of $G$ and is denoted by $γ(G)$. A vertex set $D$ is a $k$-isolating set of $G$ if $G - N_{G}[D]$ contains no $k$-cliques. The minimum cardinality of a $k$-isolating set of $G$ is called the $k$-isolation number of $G$ and is denoted by $ι_{k}(G)$. Clearly, $γ(G) = ι_{1}(G)$. A vertex set $I$ is irredundant if, for every non-isolated vertex $v$ of $G[I]$, there exists a vertex $…
We consider the so-called coupon-coloring of the vertices of a graph where every color appears in every open neighborhood, and our aim is to determine the maximal number of colors in such colorings. In other words, every color class must be a total dominating set in the graph and we study the total domatic number of the graph. We determine this parameter in every maximal outerplanar graph, and show that every Hamiltonian maximal planar graph has domatic number at least two, partially answering a conjecture of Goddard and Henning.
A set $S$ of vertices in a graph $G$ is a dominating set if every vertex not in $S$ is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The independent domination number $i(G)$ of $G$ is the minimum cardinality of an independent dominating set in $G$. In 2013 Goddard and Henning [Discrete Math 313 (2013), 839--854] conjectured that if $G$ is a connected cubic graph of order $n$, then $i(G) \le \frac{3}{8}n$, except if $G$ is the complete bipartite graph $K_{3,3}$ or the $5$-prism $C_5 \, \Box \, K_2$. Further they construct two…
In this paper, we continue the study of neighborhood total domination in graphs first studied by Arumugam and Sivagnanam [Opuscula Math. 31 (2011), 519--531]. A neighborhood total dominating set, abbreviated NTD-set, in a graph $G$ is a dominating set $S$ in $G$ with the property that the subgraph induced by the open neighborhood of the set $S$ has no isolated vertex. The neighborhood total domination number, denoted by $\gnt(G)$, is the minimum cardinality of a NTD-set of $G$. Every total dominating set is a NTD-set, implying that $γ(G) \le \gnt(G) \le \gt(G)$, where $γ(G)$ and $\gt(G)$ denot…
Let $G=(V, E)$ be a graph. A set $S\subseteq V(G)$ is a {\it dominating set} of $G$ if every vertex in $V\setminus S$ is adjacent to a vertex of $S$. The {\it domination number} of $G$, denoted by $γ(G)$, is the cardinality of a minimum dominating set of $G$. Furthermore, a dominating set $S$ is an {\it independent transversal dominating set} of $G$ if it intersects every maximum independent set of $G$. The {\it independent transversal domination number} of $G$, denoted by $γ_{it}(G)$, is the cardinality of a minimum independent transversal dominating set of $G$. In 2012, Hamid initiated the s…
In a graph $G=(V,E)$, each vertex $v\in V$ is labelled with $0$, $1$ or $2$ such that each vertex labelled with $0$ is adjacent to at least one vertex labelled $2$ or two vertices labelled $1$. Such kind of labelling is called an Italian dominating function (IDF) of $G$. The weight of an IDF $f$ is $w(f)=\sum_{v\in V}f(v)$. The Italian domination number of $G$ is $γ_{I}(G)=\min_{f} w(f)$. Gao et al. (2019) have determined the value of $γ_I(P(n,3))$. In this article, we focus on the study of the Italian domination number of generalized Petersen graphs $P(n, k)$, $k\neq3$. We determine the value…
The dominating graph of a graph G is a graph whose vertices correspond to the dominating sets of G and two vertices are adjacent whenever their corresponding dominating sets differ in exactly one vertex. Studying properties of dominating graph has become an increasingly interesting subject in domination theory. On the other hand, median graphs and partial cubes are two fundamental graph classes in graph theory. In this paper, we make some new connections between domination theory and the theory of median graphs and partial cubes. As the main result, we show that the following conditions are eq…
Given two disjoint copies of a graph $G$, denoted $G^1$ and $G^2$, and a permutation $π$ of $V(G)$, the graph $πG$ is constructed by joining $u \in V(G^1)$ to $π(u) \in V(G^2)$ for all $u \in V(G^1)$. $G$ is said to be a universal fixer if the domination number of $πG$ is equal to the domination number of $G$ for all $π$ of $V(G)$. In 1999 it was conjectured that the only universal fixers are the edgeless graphs. Since then, a few partial results have been shown. In this paper, we prove the conjecture completely.
A non-isolated vertex $x\in V(G)$ is called $C_{3}$-free if $x$ belongs to no triangle of $G$. In \cite{BMW} Burger, Mynhardt and Weakley introduced the idea of universal fixers. Let $G=(V,E)$ be a graph with $n$ vertices and $G'$ a copy of $G$. For a bijective function $π:V(G)\mapsto V (G')$, we define the prism $πG$ of $G$ as follows: $V(πG)=V(G)\cup V(G')$ and $E(πG)=E(G)\cup E(G')\cup M_π$, where $M_π=\{uπ(u): u\in V(G)\}$. Let $γ(G)$ be the domination number of $G$. If $γ(πG)=γ(G)$ for any bijective function $π$, then $G$ is called a universal fixer. In \cite{MX} it is conjectured that th…
Let $G=(V(G),E(G))$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$. Let $S$ be a subset of $V(G)$, and let $B(S)$ be the set of neighbours of $S$ in $V(G) \setminus S$. The differential $\partial(S)$ of $S$ is defined as $|B(S)|-|S|$. The maximum value of $\partial(S)$ taken over all subsets $S\subseteq V$ is the differential $\partial(G)$ of $G$. A graph operator is a mapping $F: G\rightarrow G'$, where $G$ and $G'$ are families of graphs.The graph $§{G}$ is defined as the graph obtained from $G$ con bipartición de vértices $V(G)\cup E(G)$, donde hay tantas aristas entre $v \in …
Let $G$ be a finite abelian group, written additively, and $H$ a subgroup of~$G$. The \emph{subgroup sum graph} $Γ_{G,H}$ is the graph with vertex set $G$, in which two distinct vertices $x$ and $y$ are joined if $x+y\in H\setminus\{0\}$. These graphs form a fairly large class of Cayley sum graphs. Among cases which have been considered previously are the \emph{prime sum graphs}, in the case where $H=pG$ for some prime number $p$. In this paper we present their structure and a detailed analysis of their properties. We also consider the simpler graph $Γ^+_{G,H}$, which we refer to as the \emph{…
DIENS, École normale supérieure, CNRS, PSL University（法国高等师范学院）；University of Waterloo, Department of Combinatorics and Optimization（滑铁卢大学）；Institute of Informatics, University of Warsaw（华沙大学信息学研究所）；Université de Montpellier, CNRS, LIRMM（蒙彼利埃大学）；University of Manitoba, Department of Mathematics（曼尼托巴大学）
