This page is meant to be a brief introduction to various domination problems in graph theory. Some of the applications of these problems in graph theory are in monitoring electrical networks and studying the nullity of a matrix. Only domination and zero forcing are introduced in this document. There is another variation, called power domination, that is essentially a hybrid of domination and zero forcing.
Background
Let G be a graph. The closed neighborhood of a vertex v in V, denoted N_(G)[v] or ar(N)_(G)(v), is
ar(N)_(G)(v)=N_(G)(v)cup {v},
where N_(G)(v) is the neighborhood of v.
Let G be the following graph.
Then N_(G)[v_(1)]={v_(1),v_(2),v_(4)}, which can be shown in the graph by coloring the vertices black:
If Asubseteq V, then the closed neighborhood of A is
N_(G)[A]=cup_{vin A}N_(G)[v].
Continuing the running example of the graph above, if A={v_(1),v_(6)}, then N_(G)[A]= {v_(1),v_(2),v_(4),v_(5),v_(6)} and can be depicted by coloring the vertices in the closed neighborhood:
A dominating set of a graph G=(V,E) is a subset of the vertices Dsubseteq V such that
N_(G)[D]=V.
The domination number of a graph G, denoted gamma (G), is the size of the smallest dominating set of G.
For the example above, notice that A={v_(1),v_(6)} is not a dominating set because v_(3) is not in either closed neighborhood. However, if D={v_(1),v_(5)}, then N_(G)[D]=V and D is a dominating set. Thus gamma (G)=2 (you can check by inspection that one vertex does not suffice).
To sum up the domination problem in graphs, find the smallest set of vertices such that those vertices and their neighbors color all of the vertices of the graph black.
Find gamma (G) when
(a) G=K_(n), the complete graph on n vertices.
(b) G=P_(n), the path on n vertices.
(c) G=C_(n), the cycle on n vertices.
Zero-forcing
Zero forcing is a variation of the domination problem in graphs that involves an iterative process. The goal is still the same: under this process, what is the smallest set of vertices needed to color the graph black.
The description of zero forcing is as follows.
Initialize a set of black vertices.
If a black vertex has all neighbors black except one white neighbor, color the white neighbor black.
Repeat this process until no new vertices are colored black.
The resulting black vertices are said to be forced (sometimes called infected) by the initial set. The smallest number of initial vertices needed to force the entire graph black is called the zero forcing number, Z(G). The zero forcing problem is to find the zero forcing number of a graph:
The following illustrates the zero forcing process.
(a) An initial set of black vertices
(b) v_(1) turns black since it is the only white neighbor of v_(2).
(c) v_(4) turns black since it is the only white neighbor of v_(1).
(d) v_(6) turns black since it is the only white neighbor of v_(5).
Figure 1: The forcing process
By inspection, you should get that Z(G)=3.
Find Z(G) where
(a) G=K_(n), the complete graph on n vertices.
(b) G=P_(n), the path on n vertices.
(c) G=C_(n), the cycle on n vertices.
(b) G=P_(n), the path on n vertices
(c) G=C_(n), the cycle on n vertices