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Structures of Domination in Graphs

Part of the Developments in Mathematics Book Series (volume 66)

Editors: Haynes, Teresa W.; Hedetniemi, Stephen T.; Henning, Michael A.

DOI: 10.1007/978-3-030-58892-2

ISBN: 978-3-030-58891-5

Publisher: Springer International Publishing - Springer Journals

Sep 15, 2020

1 - 10 of 17 Chapters

LitStream

Chapter 1Glossary of Common Terms

Haynes, Teresa W.; Hedetniemi, Stephen T.; Henning, Michael A.

Pages: 1 –11

[This glossary contains the most common terms and parameters of graphs that appear in this book. We include these in this introductory chapter so that they need not be redefined over and over again in each subsequent chapter.]
LitStream

Broadcast Domination in Graphs

Henning, Michael A.; MacGillivray, Gary; Yang, Feiran

Pages: 15 –46

[For a graph G, a function f : V (G) →{0, 1, …, diam(G)} is a broadcast on G, where diam(G) denotes the diameter of G. For each vertex v in G, the value f(v) is the strength of the broadcast from v. For each vertex u ∈ V (G), if there exists a vertex v in G (possibly, u = v) such that f(v) > 0...
LitStream

Alliances and Related Domination Parameters

Haynes, Teresa W.; Hedetniemi, Stephen T.

Pages: 47 –77

[Alliances in graphs were defined to model real-life applications, where an agreement exists between two or more parties to work together for the common good. In this chapter, we explore alliances and related domination parameters.]
LitStream

Fractional Domatic, Idomatic, and Total Domatic Numbers of a Graph

Goddard, Wayne; Henning, Michael A.

Pages: 79 –99

[The fractional domatic number of a graph G is the maximum ratio |ℱ|∕m(ℱ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}...
LitStream

Dominator and Total Dominator Colorings in Graphs

Henning, Michael A.

Pages: 101 –133

[A dominator coloring of a graph G is a proper coloring of the vertices of G in which each vertex of the graph dominates every vertex of some color class, where a vertex dominates itself and all vertices adjacent to it. The dominator chromatic number of G is the minimum number of colors among all...
LitStream

Irredundance

Mynhardt, C. M.; Roux, A.

Pages: 135 –181

[This chapter concerns irredundance in graphs, a concept introduced because it provides a certificate for a dominating set of a graph to be minimally dominating. Informally, a set X of vertices in a graph G is irredundant if each vertex in X dominates a vertex of G (perhaps itself) that is not...
LitStream

The Private Neighbor Concept

Hedetniemi, Stephen T.; McRae, Alice A.; Mohan, Raghuveer

Pages: 183 –218

[Given a set S ⊆ V of vertices in a graph G = (V, E), a vertex v ∈ S has a private neighbor if there is at least one vertex in the closed neighborhood N[v] of v which is not in the closed neighborhood N[S −{v}] of any vertex in S −{v}. A property P\documentclass[12pt]{minimal}...
LitStream

An Introduction to Game Domination in Graphs

Henning, Michael A.

Pages: 219 –243

[The domination game belongs to the growing family of competitive optimization graph games, and describes a process in which two players, Dominator and Staller, take turns choosing a vertex from a graph G. Each vertex chosen must dominate at least one vertex not dominated by the set of vertices...
LitStream

Domination and Spectral Graph Theory

Hoppen, Carlos; Jacobs, David P.; Trevisan, Vilmar

Pages: 245 –272

[Spectral graph theory studies graphs through the eigenvalues and eigenvectors of matrices associated with them. In this chapter we show how domination parameters have appeared in spectral graph theory, including the domination number γ, the total domination number γt, and the signed domination...
LitStream

Varieties of Roman Domination

Chellali, M.; Rad, N. Jafari; Sheikholeslami, S. M.; Volkmann, L.

Pages: 273 –307

[In this chapter, we present nine parameters, which are variations of Roman domination in graphs.]

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