scholarly journals Dominating Sequences in Grid-Like and Toroidal Graphs

10.37236/6269 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Boštjan Brešar ◽  
Csilla Bujtás ◽  
Tanja Gologranc ◽  
Sandi Klavžar ◽  
Gašper Košmrlj ◽  
...  
Keyword(s):  
Lower Bound ◽  
Domination Number ◽  
Graph Products ◽  
Lexicographic Product ◽  
Strong Product ◽  
Toroidal Graphs ◽  
Standard Graph ◽  
Domination Numbers

A longest sequence $S$ of distinct vertices of a graph $G$ such that each vertex of $S$ dominates some vertex that is not dominated by its preceding vertices, is called a Grundy dominating sequence; the length of $S$ is the Grundy domination number of $G$. In this paper we study the Grundy domination number in the four standard graph products: the Cartesian, the lexicographic, the direct, and the strong product. For each of the products we present a lower bound for the Grundy domination number which turns out to be exact for the lexicographic product and is conjectured to be exact for the strong product. In most of the cases exact Grundy domination numbers are determined for products of paths and/or cycles.

On the b-chromatic number of some graph products

2012 ◽  
Vol 49 (2) ◽  
pp. 156-169 ◽  
Author(s):  
Marko Jakovac ◽  
Iztok Peterin
Keyword(s):  
Chromatic Number ◽  
Upper And Lower Bounds ◽  
Vertex Coloring ◽  
Graph Products ◽  
Lexicographic Product ◽  
Arbitrary Graph ◽  
Strong Product ◽  
Color Class ◽  
Complete Bipartite Graphs ◽  
Proper Vertex

A b-coloring is a proper vertex coloring of a graph such that each color class contains a vertex that has a neighbor in all other color classes and the b-chromatic number is the largest integer φ(G) for which a graph has a b-coloring with φ(G) colors. We determine some upper and lower bounds for the b-chromatic number of the strong product G ⊠ H, the lexicographic product G[H] and the direct product G × H and give some exact values for products of paths, cycles, stars, and complete bipartite graphs. We also show that the b-chromatic number of Pn ⊠ H, Cn ⊠ H, Pn[H], Cn[H], and Km,n[H] can be determined for an arbitrary graph H, when integers m and n are large enough.

scholarly journals On the Laplacian spectra of product graphs

2015 ◽  
Vol 9 (1) ◽  
pp. 39-58 ◽  
Author(s):  
S. Barik ◽  
R.B. Bapat ◽  
S. Pati
Keyword(s):  
Cartesian Product ◽  
Complete Characterization ◽  
Graph Products ◽  
Lexicographic Product ◽  
Laplacian Spectrum ◽  
Strong Product ◽  
Laplacian Spectra ◽  
Product Graphs ◽  
Characteristic Sets

Graph products and their structural properties have been studied extensively by many researchers. We investigate the Laplacian eigenvalues and eigenvectors of the product graphs for the four standard products, namely, the Cartesian product, the direct product, the strong product and the lexicographic product. A complete characterization of Laplacian spectrum of the Cartesian product of two graphs has been done by Merris. We give an explicit complete characterization of the Laplacian spectrum of the lexicographic product of two graphs using the Laplacian spectra of the factors. For the other two products, we describe the complete spectrum of the product graphs in some particular cases. We supply some new results relating to the algebraic connectivity of the product graphs. We describe the characteristic sets for the Cartesian product and for the lexicographic product of two graphs. As an application we construct new classes of Laplacian integral graphs.

THE DOMINATION NUMBER OF STRONG PRODUCT OF DIRECTED CYCLES

2014 ◽  
Vol 06 (02) ◽  
pp. 1450021
Author(s):  
HUIPING CAI ◽  
JUAN LIU ◽  
LINGZHI QIAN
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Domination Number ◽  
Strong Product ◽  
Directed Cycles

Let γ(D) denote the domination number of a digraph D and let Cm ⊗ Cn denote the strong product of Cm and Cn, the directed cycles of length m, n ≥ 2. In this paper, we determine the exact values [Formula: see text] Furthermore, we give a lower bound and an upper bound of γ(Cm1 ⊗ Cm2 ⊗ ⋯ ⊗ Cmn) and obtain that [Formula: see text] when at least n-2 integers of {m1, m2, …, mn} are even (because of the isomorphism, we assume that m3, m4, …, mn are even).

scholarly journals Edge Regular Graph Products

10.37236/2817 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Boštjan Frelih ◽  
Štefko Miklavič
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Graph Products ◽  
Lexicographic Product ◽  
Normal Product ◽  
Strong Product

A regular nonempty graph $\Gamma$ is called edge regular, whenever there exists a nonegative integer $\lambda_{\Gamma}$, such that any two adjacent vertices of $\Gamma$ have precisely $\lambda_{\Gamma}$ common neighbours. An edge regular graph $\Gamma$ with at least one pair of vertices at distance 2 is called amply regular, whenever there exists a nonegative integer $\mu_{\Gamma}$, such that any two vertices at distance 2 have precisely $\mu_{\Gamma}$ common neighbours. In this paper we classify edge regular graphs, which can be obtained as a strong product, or a lexicographic product, or a deleted lexicographic product, or a co-normal product of two graphs. As a corollary we determine which of these graphs are amply regular.

scholarly journals Grundy Domination of Forests and the Strong Product Conjecture

10.37236/9507 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Kayla Bell ◽  
Keith Driscoll ◽  
Elliot Krop ◽  
Kimber Wolff
Keyword(s):  
Connected Graph ◽  
Spanning Tree ◽  
Dominating Set ◽  
Domination Number ◽  
Induced Subgraph ◽  
Strong Product ◽  
Toroidal Graphs ◽  
Maximum Sequence

A maximum sequence $S$ of vertices in a graph $G$, so that every vertex in $S$ has a neighbor which is independent, or is itself independent, from all previous vertices in $S$, is called a Grundy dominating sequence. The Grundy domination number, $\gamma_{gr}(G)$, is the length of $S$. We show that for any forest $F$, $\gamma_{gr}(F)=|V(T)|-|\mathcal{P}|$ where $\mathcal{P}$ is a minimum partition of the non-isolate vertices of $F$ into caterpillars in which if two caterpillars of $\mathcal{P}$ have an edge between them in $F$, then such an edge must be incident to a non-leaf vertex in at least one of the caterpillars. We use this result to show the strong product conjecture of B. Brešar, Cs. Bujtás, T. Gologranc, S. Klavžar, G. Košmrlj, B.~Patkós, Zs. Tuza, and M. Vizer, Dominating sequences in grid-like and toroidal graphs, Electron. J. Combin. 23(4): P4.34 (2016), for all forests. Namely, we show that for any forest $G$ and graph $H$, $\gamma_{gr}(G \boxtimes H) = \gamma_{gr}(G) \gamma_{gr}(H)$. We also show that every connected graph $G$ has a spanning tree $T$ so that $\gamma_{gr}(G)\le \gamma_{gr}(T)$ and that every non-complete connected graph contains a Grundy dominating set $S$ so that the induced subgraph of $S$ contains no isolated vertices. 

scholarly journals Lower bounds on the Roman and independent Roman domination numbers

2016 ◽  
Vol 10 (1) ◽  
pp. 65-72 ◽  
Author(s):  
Mustapha Chellali ◽  
Teresa Haynes ◽  
Stephen Hedetniemi
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Minimum Weight ◽  
Domination Number ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Roman Dominating Function ◽  
Independent Domination ◽  
Dominating Function ◽  
Domination Numbers

A Roman dominating function (RDF) on a graph G is a function f : V (G) ? {0,1,2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v of G for which f(v) = 2. The weight of a Roman dominating function is the sum f(V) = ?v?V f(v), and the minimum weight of a Roman dominating function f is the Roman domination number ?R(G). An RDF f is called an independent Roman dominating function (IRDF) if the set of vertices assigned positive values under f is independent. The independent Roman domination number iR(G) is the minimum weight of an IRDF on G. We show that for every nontrivial connected graph G with maximum degree ?, ?R(G)? ?+1/??(G) and iR(G) ? i(G) + ?(G)/?, where ?(G) and i(G) are, respectively, the domination and independent domination numbers of G. Moreover, we characterize the connected graphs attaining each lower bound. We give an additional lower bound for ?R(G) and compare our two new bounds on ?R(G) with some known lower bounds.

scholarly journals Tight Upper Bounds for the Domination Numbers of Graphs with Given Order and Minimum Degree, II

10.37236/1536 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
W. Edwin Clark ◽  
Larry A. Dunning ◽  
Stephen Suen
Keyword(s):  
Lower Bound ◽  
Minimum Degree ◽  
Domination Number ◽  
Difference Set ◽  
Upper Bounds ◽  
Circulant Graphs ◽  
Large N ◽  
The Right ◽  
Domination Numbers

Let $\gamma (n,\delta)$ denote the largest possible domination number for a graph of order $n$ and minimum degree $\delta$. This paper is concerned with the behavior of the right side of the sequence $$\gamma (n,0) \ge \gamma (n,1) \ge \cdots \ge \gamma (n,n-1) = 1. $$ We set $ \delta _k(n) = \max \{ \delta \, \vert \, \gamma (n,\delta) \ge k \}$, $k \ge 1.$ Our main result is that for any fixed $k \ge 2$ there is a constant $c_k$ such that for sufficiently large $n$, $$ n-c_kn^{(k-1)/k} \le \delta _{k+1}(n) \le n - n^{(k-1)/k}. $$ The lower bound is obtained by use of circulant graphs. We also show that for $n$ sufficiently large relative to $k$, $\gamma (n,\delta _k(n)) = k$. The case $k=3$ is examined in further detail. The existence of circulant graphs with domination number greater than 2 is related to a kind of difference set in ${\bf Z}_n$.

scholarly journals Hop Domination in Graphs-II

2015 ◽  
Vol 23 (2) ◽  
pp. 187-199
Author(s):  
C. Natarajan ◽  
S.K. Ayyaswamy
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Minimum Cardinality ◽  
Unicyclic Graphs ◽  
The Family ◽  
Independent Domination ◽  
Domination In Graphs ◽  
Domination Numbers

Abstract Let G = (V;E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V - S, there exists u ∈ S such that d(u; v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the total domination and connected domination numbers of G respectively. We then present the strong equality of hop domination and hop independent domination numbers for trees. Hop domination numbers of shadow graph and mycielskian graph of graph are also discussed.

The minus total k-domination numbers in graphs

2021 ◽  
pp. 2150150
Author(s):  
Jonecis Dayap ◽  
Nasrin Dehgardi ◽  
Leila Asgharsharghi ◽  
Seyed Mahmoud Sheikholeslami
Keyword(s):  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Sharp Bounds ◽  
Number Formula ◽  
Domination In Graphs ◽  
Dominating Function ◽  
Domination Numbers

For any integer [Formula: see text], a minus total [Formula: see text]-dominating function is a function [Formula: see text] satisfying [Formula: see text] for every [Formula: see text], where [Formula: see text]. The minimum of the values of [Formula: see text], taken over all minus total [Formula: see text]-dominating functions [Formula: see text], is called the minus total [Formula: see text]-domination number and is denoted by [Formula: see text]. In this paper, we initiate the study of minus total [Formula: see text]-domination in graphs, and we present different sharp bounds on [Formula: see text]. In addition, we determine the minus total [Formula: see text]-domination number of some classes of graphs. Some of our results are extensions of known properties of the minus total domination number [Formula: see text].

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