scholarly journals Domination Polynomials of k-Tree Related Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Somayeh Jahari ◽  
Saeid Alikhani
Keyword(s):  
Domination Number ◽  
Simple Graph ◽  
Dominating Sets ◽  
Real Roots ◽  
Domination Polynomial

Let G be a simple graph of order n. The domination polynomial of G is the polynomial DG,x=∑i=γ(G)nd(G,i)xi, 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 kinds of graphs, whose domination polynomials have no nonzero real roots.

Strong nonsplit dominating sets and strong nonsplit domination polynomial of complement of paths

2020 ◽  
Vol 12 (06) ◽  
pp. 2050082
Author(s):  
D. Kiruba Packiarani ◽  
Y. Therese Sunitha Mary
Keyword(s):  
Domination Number ◽  
Recursive Formula ◽  
Simple Graph ◽  
Dominating Sets ◽  
The Family ◽  
Domination Polynomial

Let [Formula: see text] be a simple graph of order [Formula: see text]. The strong nonsplit domination polynomial of a graph [Formula: see text] is [Formula: see text] where [Formula: see text] is the number of strong nonsplit dominating sets of [Formula: see text] of size [Formula: see text] and [Formula: see text] is the strong nonsplit domination number of [Formula: see text]. Let [Formula: see text] be the family of strong nonsplit dominating sets of a complement of a path [Formula: see text] ([Formula: see text]) with cardinality [Formula: see text], and let [Formula: see text]. In this paper, we construct [Formula: see text], the recursive formula for [Formula: see text] and [Formula: see text], the strong nonsplit domination polynomial of [Formula: see text]. Also, we obtain some properties of the coefficients of this polynomial.

scholarly journals ON THE ROOTS OF TOTAL DOMINATION POLYNOMIAL OF GRAPHS, II

2019 ◽  
pp. 659
Author(s):  
Saeid Alikhani ◽  
Nasrin Jafari
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Domination Polynomial

Let $G = (V, E)$ be a simple graph of order $n$. A  total dominating set of $G$ is a subset $D$ of $V$, such that every vertex of $V$ is adjacent to at least one vertex in  $D$. The total domination number of $G$ is  minimum cardinality of  total dominating set in $G$ and is denoted by $\gamma_t(G)$. The total domination polynomial of $G$ is the polynomial $D_t(G,x)=\sum_{i=\gamma_t(G)}^n d_t(G,i)$, where $d_t(G,i)$ is the number of total dominating sets of $G$ of size $i$. In this paper, we study roots of the total domination polynomial of some graphs.  We show that  all roots of $D_t(G, x)$ lie in the circle with center $(-1, 0)$ and radius $\sqrt[\delta]{2^n-1}$, where $\delta$ is the minimum degree of $G$. As a consequence, we prove that if $\delta\geq \frac{2n}{3}$,  then every integer root of $D_t(G, x)$ lies in the set $\{-3,-2,-1,0\}$.

scholarly journals Connected Domination Polynomial of Graphs

2018 ◽  
Vol 60 (1) ◽  
pp. 103-121 ◽  
Author(s):  
D. A. Mojdeh ◽  
A. S. Emadi
Keyword(s):  
Domination Number ◽  
Simple Graph ◽  
Dominating Sets ◽  
Connected Dominating Sets ◽  
Connected Domination Number ◽  
Domination Polynomial

Abstract Let G be a simple graph of order n. The connected domination polynomial of G is the polynomial $D_c \left( {G,x} \right) = \sum\nolimits_{i = \gamma _c \left( G \right)}^{\left| {V\left( G \right)} \right|} {d_c \left( {G,i} \right)x^i }$ , where dc(G,i) is the number of connected dominating sets of G of size i and γc(G) is the connected domination number of G. In this paper we study Dc(G,x) of any graph. We classify many families of graphs by studying their connected domination polynomial.

scholarly journals Algebraic Integers as Chromatic and Domination Roots

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Saeid Alikhani ◽  
Roslan Hasni
Keyword(s):  
Monic Polynomial ◽  
Chromatic Polynomial ◽  
Simple Graph ◽  
Dominating Sets ◽  
Algebraic Integers ◽  
Integer Coefficients ◽  
Chromatic Root ◽  
Domination Polynomial

Let G be a simple graph of order n and λ∈ℕ. A mapping f:V(G)→{1,2,…,λ} is called a λ-colouring of G if f(u)≠f(v) whenever the vertices u and v are adjacent in G. The number of distinct λ-colourings of G, denoted by P(G,λ), is called the chromatic polynomial of G. The domination polynomial of G is the polynomial D(G,λ)=∑i=1nd(G,i)λi, where d(G,i) is the number of dominating sets of G of size i. Every root of P(G,λ) and D(G,λ) is called the chromatic root and the domination root of G, respectively. Since chromatic polynomial and domination polynomial are monic polynomial with integer coefficients, its zeros are algebraic integers. This naturally raises the question: which algebraic integers can occur as zeros of chromatic and domination polynomials? In this paper, we state some properties of this kind of algebraic integers.

scholarly journals On Vertex Covering Transversal Domination Number of Regular Graphs

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
R. Vasanthi ◽  
K. Subramanian
Keyword(s):  
Upper Bound ◽  
Domination Number ◽  
Simple Graph ◽  
Regular Graphs ◽  
Dominating Sets ◽  
Cubic Graph ◽  
Minimum Cardinality ◽  
Vertex Covering

A simple graphG=(V,E)is said to ber-regular if each vertex ofGis of degreer. The vertex covering transversal domination numberγvct(G)is the minimum cardinality among all vertex covering transversal dominating sets ofG. In this paper, we analyse this parameter on different kinds of regular graphs especially forQnandH3,n. Also we provide an upper bound forγvctof a connected cubic graph of ordern≥8. Then we try to provide a more stronger relationship betweenγandγvct.

scholarly journals Neighborhood Connected k-Fair Domination Under Some Binary Operations

2019 ◽  
Vol 12 (3) ◽  
pp. 1337-1349
Author(s):  
Wardah Masanggila Bent-Usman ◽  
Rowena Isla ◽  
Sergio Canoy
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Dominating Sets ◽  
Induced Subgraph ◽  
Sharp Bounds ◽  
Binary Operations ◽  
Domination In Graphs

Let G=(V(G),E(G)) be a simple graph. A neighborhood connected k-fair dominating set (nckfd-set) is a dominating set S subset V(G) such that |N(u)  intersection S|=k for every u is an element of V(G)\S and the induced subgraph of S is connected. In this paper, we introduce and invistigate the notion of neighborhood connected k-fair domination in graphs. We also characterize such dominating sets in the join, corona, lexicographic and cartesians products of graphs and determine the exact value or sharp bounds of their corresponding neighborhood connected k-fair domination number.

scholarly journals Total Partial Domination in Graphs Under Some Binary Operations

2019 ◽  
Vol 12 (4) ◽  
pp. 1643-1655
Author(s):  
Roselainie Dimasindil Macapodi ◽  
Rowena Isla
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Dominating Sets ◽  
Cartesian Products ◽  
Sharp Bounds ◽  
Binary Operations ◽  
Domination In Graphs

Let G = (V (G), E(G)) be a simple graph and let α ∈ (0, 1]. A set S ⊆ V (G) isan α-partial dominating set in G if |N[S]| ≥ α |V (G)|. The smallest cardinality of an α-partialdominating set in G is called the α-partial domination number of G, denoted by ∂α(G). An α-partial dominating set S ⊆ V (G) is a total α-partial dominating set in G if every vertex in S isadjacent to some vertex in S. The total α-partial domination number of G, denoted by ∂T α(G), isthe smallest cardinality of a total α-partial dominating set in G. In this paper, we characterize thetotal partial dominating sets in the join, corona, lexicographic and Cartesian products of graphsand determine the exact values or sharp bounds of the corresponding total partial dominationnumber of these graphs.

The strong nonsplit domination polynomial of some graphs

2020 ◽  
Vol 12 (04) ◽  
pp. 2050053
Author(s):  
D. Kiruba Packiarani ◽  
Y. Therese Sunitha Mary
Keyword(s):  
Domination Number ◽  
Dominating Sets ◽  
Domination Polynomial ◽  
Polynomial Formula

The strong nonsplit domination polynomial of a graph [Formula: see text] of order [Formula: see text] is the polynomial [Formula: see text], where [Formula: see text] is the number of strong nonsplit dominating sets of [Formula: see text] of size [Formula: see text] and [Formula: see text] is the strong nonsplit domination number of [Formula: see text]. We obtain some properties of [Formula: see text] and its coefficients. Also, we compute the polynomial for some specific graphs.

scholarly journals On the Domination Polynomial of Some Graph Operations

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Saeid Alikhani
Keyword(s):  
Simple Graph ◽  
Dominating Sets ◽  
Graph Operations ◽  
Domination Polynomial

Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,λ)=∑i=0n‍d(G,i)λi, 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. In this paper, we study the domination polynomial of some graph operations.

More on the unimodality of domination polynomial of a graph

2021 ◽  
pp. 2150138
Author(s):  
Gee-Choon Lau ◽  
Saeid Alikhani
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Sufficient Conditions ◽  
Dominating Sets ◽  
Domination Polynomial ◽  
Polynomial Formula

Let [Formula: see text] be a graph of order [Formula: see text]. A subset [Formula: see text] of [Formula: see text] is a dominating set of [Formula: see text] if every vertex in [Formula: see text] is adjacent to at least one vertex of [Formula: see text]. The domination polynomial of [Formula: see text] is the polynomial [Formula: see text], where [Formula: see text] is the number of dominating sets of [Formula: see text] of size [Formula: see text], and [Formula: see text] is the size of a smallest dominating set of [Formula: see text], called the domination number of [Formula: see text]. Motivated by a conjecture in [S. Alikhani and Y. H. Peng, Introduction to domination polynomial of a graph, ARS Combin. 114 (2014) 257–266] which states that the domination polynomial of any graph is unimodal, we obtain sufficient conditions for this conjecture to hold. Also we study the unimodality of graph [Formula: see text] with [Formula: see text], where [Formula: see text] is an integer.

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