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 $i(G) = ι'_{1}(G)$ where $γ(G)$ and $i(G)$ are the domination and independent domination numbers. For classic results between $γ(G)$ and $i(G)$, in 1978, Allan and Laskar proved that $γ(G) = i(G)$ for all $K_{1, 3}$-free graphs and this result was generalized to $K_{1, r}$-free graphs by Bollob$\acute{a}$s and Cockayne in 1979. In 2013, Rad and Volkmann proved that the ratio $i(G)/γ(G)$ is at most $Δ(G)/2$ when $Δ(G) \in \{3, 4, 5\}$. Further, Furuya et. al. proved that when $Δ(G) \geq 6$, we have $i(G)/γ(G) \leq Δ(G) - 2\sqrt{Δ(G)} + 2$. In this paper, for a smallest $K_{k}$-isolating set $S$, we prove that $ι'_k(G)\le -\frac{ι_k^2(G)}{\ell} +i_k(G)(Δ+2)-\ell Δ$ where $\ell$ is the number of some specific vertices of $S$ such that the union of their closed neighborhoods in $S$ is $S$. We prove that this bound is sharp. A special case of our main theorem implies $ι'_{k}(G)/ι_{k}(G) \leq Δ(G) - 2\sqrt{Δ(G)} + 2$. Further, we find an inequality between $ι'_{k}(G)$ and $ι_{k}(G)$ when $G$ is $K_{1, r}$-free graph. This also generalizes the result of Bollob$\acute{a}$s and Cockayne.
We study the Nordhaus-Gaddum type results for $(k-1,k,j)$ and $k$-domination numbers of a graph $G$ and investigate these bounds for the $k$-limited packing and $k$-total limited packing numbers in graphs. As the special case $(k-1,k,j)=(1,2,0)$ we give an upper bound on $dd(G)+dd(\overline{G})$ stronger than that presented by Harary and Haynes (1996). Moreover, we establish upper bounds on the sum and product of packing and open packing numbers and characterize all graphs attaining these bounds.
A set $D$ of vertices in a graph $G$ is called dominating if every vertex of $G$ is either in $D$ or adjacent to a vertex of $D$. The paired domination number $γ_{\mathrm{pr}}(G)$ of $G$ is the minimum size of a dominating set whose induced subgraph admits a perfect matching, and the upper domination number $Γ(G)$ is the maximum size of a minimal dominating set. In this paper, we investigate the sharpness of two multiplicative inequalities for these domination parameters, where the graph product is the direct product $\times$. We show that for every positive constant $c$, there exist graphs $G…
The domination number $γ(G)$ of a graph $G$, its exponential domination number $γ_e(G)$, and its porous exponential domination number $γ_e^*(G)$ satisfy $γ_e^*(G)\leq γ_e(G)\leq γ(G)$. We contribute results about the gaps in these inequalities as well as the graphs for which some of the inequalities hold with equality. Relaxing the natural integer linear program whose optimum value is $γ_e^*(G)$, we are led to the definition of the fractional porous exponential domination number $γ_{e,f}^*(G)$ of a graph $G$. For a subcubic tree $T$ of order $n$, we show $γ_{e,f}^*(T)=\frac{n+2}{6}$ and $γ_e(T…
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.
TxGraffiti is an automated conjecturing program that produces graph theoretic conjectures in the form of conjectured inequalities. This program written and maintained by the second author since 2017 was inspired by the successes of previous automated conjecturing programs including Fajtlowicz's GRAFFITI and DeLaViña's GRAFFITI.pc. In this paper we prove and generalize several conjectures generated by TxGraffiti when it was prompted to conjecture on the \emph{independence number}, the \emph{domination number}, and the \emph{matching number} (and generalizations of each of these graph invariants…
A $k$-$γ_{c}$-edge critical graph is a graph $G$ with the connected domination number $γ_{c}(G) = k$ and $γ_{c}(G + uv) < k$ for every $uv \in E(\overline{G})$. Further, a $2$-connected graph $G$ is said to be $k$-$γ_{c}$-vertex critical if $γ_{c}(G) = k$ and $γ_{c}(G - v) < k$ for all $v \in V(G)$. A maximal $k$-$γ_{c}$-vertex critical graph is a graph which are both $k$-$γ_{c}$-edge critical and $k$-$γ_{c}$-vertex critical. Let $κ, δ, ω$ and $α$ be respectively connectivity minimum degree, clique number and independence number. In this paper, we prove that every maximal $3$-$γ_{c}$-vertex cr…
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 $…
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.
In this paper we study the weak Roman domination number and the secure domination number of a graph. In particular, we obtain general bounds on these two parameters and, as a consequence of the study, we derive new inequalities of Nordhaus-Gaddum type involving secure domination and weak Roman domination. Furthermore, the particular case of Cartesian product graphs is considered.
Nine variations of the concept of domination in a simple graph are identified as fundamental domination concepts, and a unified approach is introduced for studying them. For each variation, the minimum cardinality of a subset of dominating elements is the corresponding fundamental domination number. It is observed that, for each nontrivial connected graph, at most five of these nine numbers can be different, and inequalities between these five numbers are given. Finally, these fundamental dominations are interpreted in terms of the total graph of the given graph, a concept introduced by the se…
Two inequalities bridging the three isolated graph invariants, incidence chromatic number, star arboricity and domination number, were established. Consequently, we deduced an upper bound and a lower bound of the incidence chromatic number for all graphs. Using these bounds, we further reduced the upper bound of the incidence chromatic number of planar graphs and showed that cubic graphs with orders not divisible by four are not 4-incidence colorable. The incidence chromatic numbers of Cartesian product, join and union of graphs were also determined.
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…
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 so…
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 $k-$quasiperfect dominating set ($k\ge 1$) of a graph $G$ is a vertex subset $S$ such that every vertex not in $S$ is adjacent to at least one and at most k vertices in $S$. The cardinality of a minimum k-quasiperfect dominating set in $G$ is denoted by $γ_{\stackrel{}{1k}}(G)$. Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept. The quasiperfect domination chain $γ_{\stackrel{}{11}}(G)\geγ_{\stackrel{}{12}}(G)\ge\dots\geγ_{\stackrel{}{1Δ}}(G)=γ(G)$, indicates what it is lost in size when you move towards a more perfect domination…
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…
Imagine that we are given a set $D$ of officials and a set $W$ of civils. For each civil $x \in W$, there must be an official $v \in D$ that can serve $x$, and whenever any such $v$ is serving $x$, there must also be another civil $w \in W$ that observes $v$, that is, $w$ may act as a kind of witness, to avoid any abuse from $v$. What is the minimum number of officials to guarantee such a service, assuming a given social network? In this paper, we introduce the concept of certified domination that perfectly models the aforementioned problem. Specifically, a dominating set $D$ of a graph $G=(V_…
Let $G$ be a graph and let $S\subseteq V(G)$. The set $S$ is a double outer-independent dominating set of $G$ if $|N[v]\cap D|\geq2$, for all $v\in V(G)$, and $V(G)\setminus S$ is independent. Similarly, $S$ is a $2$-outer-independent dominating set, if every vertex from $V(G)\setminus S$ has at least two neighbors in $S$ and $V(G)\setminus S$ is independent. Finally, $S$ is a total outer-independent dominating set if every vertex from $V(G)$ has a neighbor in $S$ and the complement of $S$ is an independent set. The double, total or $2$-outer-independent domination number of $G$ is the smalles…
Rubalcaba and Slater (Robert R. Rubalcaba and Peter J. Slater. Efficient (j,k)-domination. Discuss. Math. Graph Theory, 27(3):409-423, 2007.) define a $(j,k)$-dominating function on graph $X$ as a function $f:V(X)\rightarrow \{0,\ldots,j\}$ so that for each $v\in V(X)$, $f(N[v])\geq k$, where $N[v]$ is the closed neighbourhood of $v$. Such a function is efficient if all of the vertex inequalities are met with equality. They give a simple necessary condition for efficient domination, namely: if $X$ is an $r$-regular graph on $n$ vertices that has an efficient $(1,k)$-dominating function, then 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.
Let $m_G(I)$ denote the number of Laplacian eigenvalues of a graph $G$ in an interval $I$, and let $γ(G)$ denote its domination number. We extend the recent result $m_G[0,1) \leq γ(G)$, and show that isolate-free graphs also satisfy $γ(G) \leq m_G[2,n]$. In pursuit of better understanding Laplacian eigenvalue distribution, we find applications for these inequalities. We relate these spectral parameters with the approximability of $γ(G)$, showing that $\frac{γ(G)}{m_G[0,1)} \not\in O(\log n)$. However, $γ(G) \leq m_G[2, n] \leq (c + 1) γ(G)$ for $c$-cyclic graphs, $c \geq 1$. For trees $T$, $γ(…
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…
The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy; one uses Fuglede-Kadison determinants, while another uses effective resistance. We use the latter to prove that tree entropy respects stochastic domination. We also prove that tree entropy is non-negative in the unweighted case, a special case of which establishes Lueck's Determinant Conjecture for Cayley-graph Laplacians. We use techniques from the theory of operators affi…
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.
We prove a conjecture of Postnikov, Reiner and Williams by defining a partial order on the set of tree graphs with $n$ vertices that induces inequalities between the $γ$-polynomials of their associated graph-associahedra. The partial order is given by relating trees that can be obtained from one another by operations called tree shifts. We also show that tree shifts lower the $γ$-polynomials of graphs that are not trees, as do the flossing moves of Babson and Reiner.
For a graph $G= (V,E)$, a double Roman dominating function (DRDF) is a function $f : V \to \{0,1,2,3\}$ having the property that if $f (v) = 0$, then vertex $v$ must have at least two neighbors assigned $2$ under $f$ or {at least} one neighbor $u$ with $f (u) = 3$, and if $f (v) = 1$, then vertex $v$ must have at least one neighbor $u$ with $f (u) \ge 2$. In this paper, we consider the double Roman domination problem, which is an optimization problem of finding the DRDF $f$ such that $\sum_{v\in V} f (v)$ is minimum. We propose {five integer linear programming (ILP) formulations and one mixed …
