The isodominism class of the graphs

2020 ◽  
pp. 2150030
Author(s):  
Behnaz Tolue ◽  
Amin Rafiei
Keyword(s):  
Positive Integer ◽  
Graph Isomorphism ◽  
Complete Graphs ◽  
Dominating Sets

In this paper, the new concept [Formula: see text]-isodominism between two graphs is introduced, where [Formula: see text] is a positive integer. The [Formula: see text]-isodominism is a pair of the graph isomorphism which depends on the smallest [Formula: see text]-dominating sets. By this tool, a graph with one vertex, complete graphs, wheels and stars are in the same [Formula: see text]-isodominism class. Furthermore, some results about the [Formula: see text]-isodominism classes of paths, cycles and star polygon are presented.

The minimum eccentric distance sum of trees with given distance k-domination number

2020 ◽  
Vol 12 (04) ◽  
pp. 2050052 ◽  
Author(s):  
Lidan Pei ◽  
Xiangfeng Pan
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Number Formula ◽  
Simple Connected Graph ◽  
Eccentric Distance ◽  
Distance Formula

Let [Formula: see text] be a positive integer and [Formula: see text] be a simple connected graph. The eccentric distance sum of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the maximum distance from [Formula: see text] to any other vertex and [Formula: see text] is the sum of all distances from [Formula: see text]. A set [Formula: see text] is a distance [Formula: see text]-dominating set of [Formula: see text] if for every vertex [Formula: see text], [Formula: see text] for some vertex [Formula: see text]. The minimum cardinality among all distance [Formula: see text]-dominating sets of [Formula: see text] is called the distance [Formula: see text]-domination number [Formula: see text] of [Formula: see text]. In this paper, the trees among all [Formula: see text]-vertex trees with distance [Formula: see text]-domination number [Formula: see text] having the minimal eccentric distance sum are determined.

Dominating a Family of Graphs with Small Connected Subgraphs

2000 ◽  
Vol 9 (4) ◽  
pp. 309-313 ◽  
Author(s):  
YAIR CARO ◽  
RAPHAEL YUSTER
Keyword(s):  
Positive Integer ◽  
Minimum Degree ◽  
Minimum Size ◽  
Dominating Sets ◽  
Asymptotic Results ◽  
Connected Dominating Sets ◽  
Vertex Set ◽  
Connected Subgraphs

Let F = {G1, …, Gt} be a family of n-vertex graphs defined on the same vertex-set V, and let k be a positive integer. A subset of vertices D ⊂ V is called an (F, k)-core if, for each v ∈ V and for each i = 1, …, t, there are at least k neighbours of v in Gi that belong to D. The subset D is called a connected (F, k)-core if the subgraph induced by D in each Gi is connected. Let δi be the minimum degree of Gi and let δ(F) = minti=1δi. Clearly, an (F, k)-core exists if and only if δ(F) [ges ] k, and a connected (F, k)-core exists if and only if δ(F) [ges ] k and each Gi is connected. Let c(k, F) and cc(k, F) be the minimum size of an (F, k)-core and a connected (F, k)-core, respectively. The following asymptotic results are proved for every t < ln ln δ and k < √lnδ:formula hereThe results are asymptotically tight for infinitely many families F. The results unify and extend related results on dominating sets, strong dominating sets and connected dominating sets.

scholarly journals On Symmetry of Complete Graphs over Quadratic and Cubic Residues

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
M. Haris Mateen ◽  
M. Khalid Mahmood ◽  
Shahbaz Ali ◽  
M. D. Ashraful Alam
Keyword(s):  
Positive Integer ◽  
Quadratic Residue ◽  
The Other ◽  
Complete Graphs

In this study, we investigate two graphs, one of which has units of a ring Z n as vertices (or nodes) and an edge will be built between two vertices u and v if and only if u 3 ≡ v 3 mod   n . This graph will be termed as cubic residue graph. While the other is called Gaussian quadratic residue graph whose vertices are the elements of a Gaussian ring Z n i of the form α = a + i b , β = c + i    d , where a , b , c , d are the units of Z n . Two vertices α and β are adjacent to each other if and only if α 2 ≡ β 2 mod   n . In this piece of work, we characterize cubic and Gaussian quadratic residue graphs for each positive integer n in terms of complete graphs.

scholarly journals On 1-rotational decompositions of complete graphs into tripartite graphs

2019 ◽  
Vol 39 (5) ◽  
pp. 623-643
Author(s):  
Ryan C. Bunge
Keyword(s):  
Positive Integer ◽  
Simple Graph ◽  
Vertex Coloring ◽  
Complete Graphs ◽  
Tripartite Graph ◽  
Tripartite Graphs ◽  
Proper Vertex

Consider a tripartite graph to be any simple graph that admits a proper vertex coloring in at most 3 colors. Let \(G\) be a tripartite graph with \(n\) edges, one of which is a pendent edge. This paper introduces a labeling on such a graph \(G\) used to achieve 1-rotational \(G\)-decompositions of \(K_{2nt}\) for any positive integer \(t\). It is also shown that if \(G\) with a pendent edge is the result of adding an edge to a path on \(n\) vertices, then \(G\) admits such a labeling.

scholarly journals General Form of Domination Polynomial for Two Types of Graphs Associated to Dihedral Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (2) ◽  
pp. 149-155
Author(s):  
Nabilah Najmuddin ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Conjugacy Classes ◽  
Complete Graphs ◽  
Dominating Sets ◽  
Dihedral Groups ◽  
Graph Polynomial ◽  
Central Elements ◽  
Domination Polynomial ◽  
Class Graph

A domination polynomial is a type of graph polynomial in which its coefficients represent the number of dominating sets in the graph. There are many researches being done on the domination polynomial of some common types of graphs but not yet for graphs associated to finite groups. Two types of graphs associated to finite groups are the conjugate graph and the conjugacy class graph. A graph of a group G is called a conjugate graph if the vertices are non-central elements of G and two distinct vertices are adjacent if they are conjugate to each other. Meanwhile, a conjugacy class graph of a group G is a graph in which its vertices are the non-central conjugacy classes of G and two distinct vertices are connected if and only if their class cardinalities are not coprime. The conjugate and conjugacy class graph of dihedral groups can be expressed generally as a union of complete graphs on some vertices. In this paper, the domination polynomials are computed for the conjugate and conjugacy class graphs of the dihedral groups.

scholarly journals Packing chromatic numbers of finite super subdivisions of graphs

Filomat ◽  
2020 ◽  
Vol 34 (10) ◽  
pp. 3275-3286
Author(s):  
Rachid Lemdani ◽  
Moncef Abbas ◽  
Jasmina Ferme
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
Complete Graphs ◽  
Chromatic Numbers ◽  
General Properties ◽  
Vertex Set ◽  
Packing Chromatic Number ◽  
Corona Graphs

Given a graph G and a positive integer i, an i-packing in G is a subset W of the vertex set of G such that the distance between any two distinct vertices from W is greater than i. The packing chromatic number of a graph G, ??(G), is the smallest integer k such that the vertex set of G can be partitioned into sets Vi, i ? {1,..., k}, where each Vi is an i-packing. In this paper, we present some general properties of packing chromatic numbers of finite super subdivisions of graphs. We determine the packing chromatic numbers of the finite super subdivisions of complete graphs, cycles and some neighborhood corona graphs.

scholarly journals Perfect and quasiperfect domination in trees

2016 ◽  
Vol 10 (1) ◽  
pp. 46-64
Author(s):  
José Cáceres ◽  
Carmen Hernando ◽  
Mercè Mora ◽  
Ignacio Pelayo ◽  
María Puertas
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Linear Algorithm ◽  
G 12 ◽  
Graph Parameters

A k??quasiperfect dominating set of a connected 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 ?1k(G). These graph parameters were first introduced by Chellali et al. (2013) as a generalization of both the perfect domination number ?11(G) and the domination number ?(G). The study of the so-called quasiperfect domination chain ?11(G) ? ?12(G)?... ? ?1?(G) = ?(G) enable us to analyze how far minimum dominating sets are from being perfect. In this paper, we provide, for any tree T and any positive integer k, a tight upper bound of ?1k(T). We also prove that there are trees satisfying all possible equalities and inequalities in this chain. Finally a linear algorithm for computing ?1k(T) in any tree T is presented.

Weak dominating sets and Weak Domination Polynomial of Complete Graphs

2018 ◽  
Vol 9 (12) ◽  
pp. 2138-2146
Author(s):  
S. Angelin Kavitha Raj ◽  
S. Robinson Chelladurai
Keyword(s):  
Complete Graphs ◽  
Dominating Sets ◽  
Domination Polynomial

Bounds on the k-tuple domatic number of a graph

2011 ◽  
Vol 61 (6) ◽  
Author(s):  
Lutz Volkmann
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Dominating Set ◽  
Simple Graph ◽  
Dominating Sets ◽  
Domatic Number ◽  
Vertex Set

AbstractLet k be a positive integer, and let G be a simple graph with vertex set V (G). A vertex of a graph G dominates itself and all vertices adjacent to it. A subset S ⊆ V (G) is a k-tuple dominating set of G if each vertex of V (G) is dominated by at least k vertices in S. The k-tuple domatic number of G is the largest number of sets in a partition of V (G) into k-tuple dominating sets.In this paper, we present a lower bound on the k-tuple domatic number, and we establish Nordhaus-Gaddum inequalities. Some of our results extends those for the classical domatic number.

Critical Exponents, Colines, and Projective Geometries

2000 ◽  
Vol 9 (4) ◽  
pp. 355-362 ◽  
Author(s):  
JOSEPH P. S. KUNG
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Dimensional Subspace ◽  
Complete Graphs ◽  
Matroid Theory ◽  
Primary Interest ◽  
Projective Geometries ◽  
Critical Problems ◽  
Simple Matroid ◽  
Geometric Analogue

In [9, p. 469], Oxley made the following conjecture, which is a geometric analogue of a conjecture of Lovász (see [1, p. 290]) about complete graphs.Conjecture 1.1.Let G be a rank-n GF(q)-representable simple matroid with critical exponent n − γ. If, for every coline X in G, c(G/X; q) = c(G; q) − 2 = n − γ − 2, then G is the projective geometry PG(n − 1, q).We shall call the rank n, the critical ‘co-exponent’ γ, and the order q of the field the parameters of Oxley's conjecture. We exhibit several counterexamples to this conjecture. These examples show that, for a given prime power q and a given positive integer γ, Oxley's conjecture holds for only finitely many ranks n. We shall assume familiarity with matroid theory and, in particular, the theory of critical problems. See [6] and [9].A subset C of points of PG(n − 1, q) is a (γ, k)-cordon if, for every k-codimensional subspace X in PG(n − 1, q), the intersection C ∩ X contains a γ-dimensional subspace of PG(n − 1, q). In this paper, our primary interest will be in constructing (γ, 2)-cordons. With straightforward modifications, our methods will also yield (γ, k)-cordons.Complements of counterexamples to Oxley's conjecture are (γ, 2)-cordons.

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