Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x)=\sum_{i=γ(G)}^{n} d(G,i) x^{i}, where d(G,i) is the number of dominating sets of G of size i, and γ(G) is the domination number of G. In this paper we study the domination polynomials of cubic graphs of order 10. As a consequence, we show that the Petersen graph is determined uniquely by its domination polynomial.
Let G be a simple undirected graph with no isolated vertex. A paired dominating set of G is a dominating set which induces a subgraph that has a perfect matching. The paired domination number of G, denoted by γpr(G), is the size of its smallest paired dominating set. Goddard and Henning conjectured that γpr(G) {\leq} 4n/7 holds for every graph G with δ(G) {\geq} 3, except the Petersen Graph. In this paper, we prove this conjecture for cubic graphs.
The power domination problem focuses on finding the optimal placement of phase measurement units (PMUs) to monitor an electrical power network. In the context of graphs, the power domination number of a graph $G$, denoted $γ_P(G)$, is the minimum number of vertices needed to observe every vertex in the graph according to a specific set of observation rules. In \cite{ZKC_cubic}, Zhao et al. proved that if $G$ is a connected claw-free cubic graph of order $n$, then $γ_P(G) \leq n/4$. In this paper, we show that if $G$ is a claw-free diamond-free cubic graph of order $n$, then $γ_P(G) \le n/6$, a…
Let $G$ be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x) =\sum d(G, i)x^i, where d(G,i) is the number of dominating sets of G of size i. Every root of D(G,x) is called the domination root of G. It is clear that (0,\infty) is zero free interval for domination polynomial of a graph. It is interesting to investigate graphs which have complex domination roots with positive real parts. In this paper, we first investigate complexity of the domination polynomial at specific points. Then we present and investigate some families of graphs whose complex domination ro…
Let v(G) and dom(G) denote the number of vertices and the domination number of a graph G, and let r (G) = dom(G)/v(G)$. Let [x] and ]x[ be the floor and the ceiling of a number x. In 1996 B. Reed conjectured that if G is a cubic graph, then dom(G) is at most ]v(G)/3[. In 2005 A. Kostochka and B. Stodolsky disproved this conjecture for cubic graphs of connectivity one and maintained that the conjecture may still be true for 2-connected cubic graphs. Their minimum counterexample C has 4 bridges, v(C) = 60, anddom (C) = 21. In this paper we disprove Reed's conjecture for 2-connected cubic graphs …
Let $G$ be a graph on $n$ vertices and $m$ edges and $D(G,x)$ the domination polynomial of $G$. In this paper we completely characterize the values of $n$ and $m$ for which optimal graphs exist for domination polynomials. We also show that there does not always exist least optimal graphs for the domination polynomial. Applications to network reliability are highlighted.
A polynomial is said to be unimodal if its coefficients are non-decreasing and then non-increasing. The domination polynomial of a graph $G$ is the generating function of the number of domination sets of each cardinality in $G$, and its coefficients have been conjectured to be unimodal. In this paper we will show the domination polynomial of paths, cycles and complete multipartite graphs are unimodal, and that the domination polynomial of almost every graph is unimodal with mode $ \lceil \frac{n}{2}\rceil $.
Given a simple, finite graph with vertex set $V(G)$, we define a zero forcing set of $G$ as follows. Choose $S\subseteq V(G)$ and color all vertices of $S$ blue and all vertices in $V(G) - S$ white. The color change rule is if $w$ is the only white neighbor of blue vertex $v$, then we change the color of $w$ from white to blue. If after applying the color change rule as many times as possible eventually every vertex of $G$ is blue, we call $S$ a zero forcing set of $G$. $Z(G)$ denotes the minimum cardinality of a zero forcing set. Davila and Henning proved in \cite{zerocubic} that for any claw…
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.
Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=γ(G)}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$ and $γ(G)$ is the domination number of $G$. In this paper we study the domination polynomials of several classes of $k$-tree related graphs. Also, we present families of these kind of graphs, whose domination polynomial have no nonzero real roots.
A dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex outside $D$ is adjacent to a vertex in $D$. A locating-dominating set of $G$ is a dominating set $D$ of $G$ with the additional property that every two distinct vertices outside $D$ have distinct neighbors in $D$; that is, for distinct vertices $u$ and $v$ outside $D$, $N(u) \cap D \neq N(v) \cap D$ where $N(u)$ denotes the open neighborhood of $u$. A graph is twin-free if every two distinct vertices have distinct open and closed neighborhoods. The location-domination number of $G$, denoted $γ_L(G)$, is the …
Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=γ(G)}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$ and $γ(G)$ is the domination number of $G$. In this paper we consider cactus chains with triangular and square blocks and study their domination polynomials.
Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G,λ)=\sum_{i=0}^{n} d(G,i) λ^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. We consider the lexicographic product of two specific graphs and study their domination polynomials.
Let G be a simple graph of order n. The domination polynomial of a graph is the generating function of its dominating sets. We study the domination polynomials of generalized friendship graphs. We also consider book graphs formed by joining n copies of the cycle graph of order 4 with a common edge and study the domination polynomials of some generalized book graphs. In particular we examine the domination roots of these families and find the limiting curve for the roots.
The domination polynomial of a graph $G$ of order $n$ is the polynomial $D(G,x)=\sum_{i=γ(G)}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$, and $γ(G)$ is the domination number of $G$. The roots of domination polynomial is called domination roots. In this article, we compute the domination polynomial and domination roots of all graphs of order less than or equal to 6, and show them in the tables.
Given a graph $G$, a total dominating set $D_t$ is a vertex set that every vertex of $G$ is adjacent to some vertices of $D_t$ and let $d_t(G,i)$ be the number of all total dominating sets with size $i$. The total domination polynomial, defined as $D_t(G,x)=\sum\limits_{i=1}^{| V(G)|} d_t(G,i)x^i$, recently has been one of the considerable extended research in the field of domination theory. In this paper, we obtain the vertex-reduction and edge-reduction formulas of total domination polynomials. As consequences, we give the total domination polynomials for paths and cycles. Additionally, we d…
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 properties of D(G, x) and its coefficients. Also we compute this polynomial for some specific graphs.
We introduce a new bivariate polynomial ${\displaystyle J(G; x,y):=\sum\limits_{W \in V(G)} x^{|W|}y^{|N(W)|}}$ which contains the standard domination polynomial of the graph $G$ in two different ways. We build methods for efficient calculation of this polynomial and prove that there are still some families of graphs which have the same bivariate polynomial.
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 domination polynomial of a graph $G$ of order $n$, defined as $\mathcal{P}(G;x)=\sum_{i=1}^n p(G;i) x^i$, where $p(G;i)$ is the number of power dominating sets of $G$ of size $i$. We relate the power domination polynomial to other graph polynomials, present structural and extremal results about its roots and coefficients, and identify some graph parameters …
The domination polynomial of a graph is the polynomial whose coefficients count the number of dominating sets of each cardinality. A recent question asks which graphs are uniquely determined (up to isomorphism) by their domination polynomial. In this paper, we completely describe the complete r-partite graphs which are; in the bipartite case, this settles in the affirmative a conjecture of Aalipour, Akbari and Ebrahimi.
An independent dominating set of the simple graph $G=(V,E)$ is a vertex subset that is both dominating and independent in $G$. The independent domination polynomial of a graph $G$ is the polynomial $D_i(G,x)=\sum_{A} x^{|A|}$, summed over all independent dominating subsets $A\subseteq V$. A root of $D_i(G,x)$ is called an independence domination root. We investigate the independent domination polynomials of some generalized compound graphs. As consequences, we construct graphs whose independence domination roots are real. Also, we consider some certain graphs and study the number of their inde…
A vertex subset $W\subseteq V$ of the graph $G=(V,E)$ is an independent dominating set if every vertex in $V\backslash W$ is adjacent to at least one vertex in $W$ and the vertices of $W$ are pairwise non-adjacent. The independent domination polynomial is the ordinary generating function for the number of independent dominating sets in the graph. We investigate in this paper properties of the independent domination polynomial and some interesting connections to well known counting problems.
Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=\sum_{i=γ(G)}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$ and $γ(G)$ is the domination number of $G$. It is conjectured that the domination polynomial of any graph is unimodal. In this paper we present some families of graphs whose domination polynomials are unimodal.
A dominating set $S$ of a graph $G$ of order $n$ is a subset of the vertices of $G$ such that every vertex is either in $S$ or adjacent to a vertex of $S$. The domination polynomial is defined by $D(G,x) = \sum d_k x^k$ where $d_k$ is the number of dominating sets in $G$ with cardinality $k$. In this paper we show that the closure of the real roots of domination polynomials is $(-\infty,0]$.
Let $G$ be a simple graph of order $n$. A dominating set of $G$ is a set $S$ of vertices of $G$ so that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. The domination polynomial of $G$ 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 $i$. In this paper we show that cycles are determined by their domination polynomials.
Given a finite graph $G$, the maximum length of a sequence $(v_1,\ldots,v_k)$ of vertices in $G$ such that each $v_i$ dominates a vertex that is not dominated by any vertex in $\{v_1,\ldots,v_{i-1}\}$ is called the Grundy domination number, $γ_{\rm gr}(G)$, of $G$. A small modification of the definition yields the Z-Grundy domination number, which is the dual invariant of the well-known zero forcing number. In this paper, we prove that $γ_{\rm gr}(G) \geq \frac{n + \lceil \frac{k}{2} \rceil - 2}{k-1}$ holds for every connected $k$-regular graph of order $n$ different from $K_{k+1}$ and $\bar{2…
A subset $D$ of vertices of a graph $G$ is a \textit{dominating set} if for each $u\in V(G)\setminus D$, $u$ is adjacent to some vertex $v\in D$. The \textit{dominating number}, $γ(G)$ of $G$, is the minimum cardinality of a dominating set of $G$. A set $D\subseteq V(G)$ is a \textit{total dominating set} if for each $u\in V(G)$, $u$ is adjacent to some vertex $v\in D$. the The \textit{total dominating number}, $γ_t(G)$ of $G$, is the minimum cardinality of a total dominating set of $G$. For an even integer $n\ge2$ and $1\leΔ\le\lfloor\log_2n\rfloor$, a \textit{Knödel graph} $W_{Δ,n}$ is a $Δ$…
Let $G$ be a simple graph of order $n$. The {\em domination polynomial} of $G$ is the polynomial ${D(G, x)=\sum_{i=0}^{n} d(G,i) x^{i}}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. Let $n$ be any positive integer and $F_n$ be the Friendship graph with $2n + 1$ vertices and $3n$ edges, formed by the join of $K_{1}$ with $nK_{2}$. We study the domination polynomials of this family of graphs, and in particular examine the domination roots of the family, and find the limiting curve for the roots. We also show that for every $n\geq 2$, $F_n$ is not $\mathcal{D}$-unique, tha…
Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G,λ)=\sum_{i=0}^{n} d(G,i) λ^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. Every root of $D(G,λ)$ is called the domination root of $G$. We present families of graphs whose their domination polynomial have no nonzero real roots. We observe that these graphs have complex domination roots with positive real part. Then, we consider the lexicographic product of two graphs and obtain a formula for domination polynomial of this product. Using this product, we construct a family of…
From a research of several recent papers, in the first part, we are concerned with domination number in cubic graphs and give a sufficient condition of Reed's conjecture. In the second part, from a perspective, we study the structure of a minimum dominating set in 3-connected graphs. It is derived from a collection of cycles with length 0 mod 3.
We propose the conjecture that the domination number $γ(G)$ of a $Δ$-regular graph $G$ with $Δ\geq 1$ is always at most its edge domination number $γ_e(G)$, which coincides with the domination number of its line graph. We prove that $γ(G)\leq \left(1+\frac{2(Δ-1)}{Δ2^Δ}\right)γ_e(G)$ for general $Δ\geq 1$, and $γ(G)\leq \left(\frac{7}{6}-\frac{1}{204}\right)γ_e(G)$ for $Δ=3$. Furthermore, we verify our conjecture for cubic claw-free graphs.
Let G be a simple graph of order n. The domination polynomial is the generating polynomial for the number of dominating sets of G of each cardinality. A root of this polynomial is called a domination root of G. Obviously 0 is a domination root of every graph G. In the study of the domination roots of graphs, this naturally raises the question: which graphs have no nonzero real domination roots? In this paper we present some families of graphs whose have this property.
The $k$-rainbow domination problem is studied for regular graphs. We prove that the $k$-rainbow domination number $γ_{rk}(G)$ of a $d$-regular graph for $d\leq k\leq 2d$ is bounded below by $\displaystyle{\left\lceil kn/2d\right\rceil}$, where $n$ is the order of a graph. We determine necessary conditions for regular graphs to attain this bound and find several examples. As an application, we determine exact $k$-rainbow domination numbers for all cubic Cayley graphs over abelian groups.
The domination polynomial D(G,x) of a graph G is the generating function of its dominating sets. We prove that D(G,x) satisfies a wide range of reduction formulas. We show linear recurrence relations for D(G,x) for arbitrary graphs and for various special cases. We give splitting formulas for D(G,x) based on articulation vertices, and more generally, on splitting sets of vertices.
Counterexamples to two conjectures on (1, 2)-domination in cubic graphs
Slovak University of Technology in Bratislava（布拉迪斯拉发斯洛伐克理工大学）；University of Split（斯普利特大学）；University of Novo Mesto（新梅斯托大学）；Rudolfovo – Science and Technology Centre Novo Mesto（鲁道尔福沃-新梅斯托科学与技术中心）；University of Ljubljana（卢布尔雅那大学）
Given a simple graph $G$ on $n$ vertices, a subset of vertices $U \subseteq V(G)$ is dominating if every vertex of $V(G)$ is either in $U$ or adjacent to a vertex of $U$. The domination polynomial of $G$ is the generating function whose coefficients are the number of dominating sets of a given size. We show that the domination polynomial is unimodal, i.e., the coefficients are non-decreasing and then non-increasing, for several well-known families of graphs. In particular, we prove unimodality for spider graphs with at most $400$ legs (of arbitrary length), lollipop graphs, arbitrary direct pr…
Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set of $G$ is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and denoted by $γ_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=γ_t(G)}^n d_t(G,i)$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. In this paper, we study roots of total domination polynomial of some graphs. We show that all roots of $D_t(G, x)$ lie in the circle with c…
Let G=(V,E) be a simple graph. A set S\subset V is a dominating set of G, if every vertex in V§is adjacent to at least one vertex in S. Let {\mathcal C}_n^i be the family of dominating sets of a cycle C_n with cardinality i, and let d(C_n,i) = |{\mathcal C}_n^i. In this paper, we construct {\mathcal C}_n^i, and obtain a recursive formula for d(C_n, i). Using this recursive formula, we consider the polynomial D(C_n, x) = \sum_{i=1}^n d(C_n, i)x^i, which we call domination polynomial of cycles and obtain some properties of this polynomial.
Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set of $G$ is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and is denoted by $γ_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=γ_t(G)}^n d_t(G,i)x^i$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. Two graphs $G$ and $H$ are said to be total dominating equivalent or simply $\mathcal{D}_t$-equivalent, if $D_t(G,x)=D_t(H,x)$. The e…
Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set of $G$ is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of total dominating set in $G$ and is denoted by $γ_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=γ_t(G)}^n d_t(G,i)x^i$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. A root of $D_t(G,x)$ is called a total domination root of $G$. An irrelevant edge of $D_t(G,x)$ is an edge $e \in E$, such that $D_t(…
