The first step to obtain the desired partition is the introduction of twins. As we will see, it provides a helpful frame when dealing with equidomination. It is also used in the context of equistability (see for example [38]).
Definition 4.3.1. Let G = (V, E) be a graph. Two vertices v, w ∈ V are called twins if they have the same neighborhood except themselves, that is if
N (v) \ {w} = N (w) \ {v} .
If further vw ∈ E, we say v and w are true twins and otherwise false twins. We define a relation using the twin property:
v ∼tw :⇐⇒ N (v) \ {w} = N (w) \ {v} .
This relation will be referred to as the twin relation.
Since ∼t-related vertices dominate the same set of vertices (besides themselves in the
case of false twins), it seems reasonable to work with the twin relation in the context of (minimal) dominating sets.
Lemma 4.3.2. The twin relation is an equivalence relation.
Proof. Let G be a graph with vertex set V and edge set E. Symmetry and reflexivity follow immediately from the definition. For transitivity, let v, w and x ∈ V with v ∼tw
and w ∼t x. Then N (v) \ {w, x} = N (w) \ {v, x} = N (x) \ {v, w} and for symmetry
reasons it remains to show that vw ∈ E if wx ∈ E. So let wx ∈ E. Since v and w are related, we get vw ∈ E and the proof is finished.
4.3 Decomposition a b c1 c2 d e s1 s2 16 7 3 3 3 4 8 8
Figure 4.1: An equidominating graph on 8 vertices; the target value is t = 23; its twin partition is {a}, {b}, {c1, c2}, {d}, {e}, {s1, s2}
and its pseudo class partition
{a}, {b}, {c1, c2, d}, {e}, {s1, s2}
.
Since the considered partition is based on an equivalence relation – more precisely on equivalence classes – we will use the term class instead of blocks in the following. The equivalence classes of ∼t are called twin classes and the partition of the vertices into
twin classes is called twin partition. Similar to the proof of Lemma 4.3.2 one can show that all vertices of a twin class are either pairwise adjacent or pairwise non-adjacent. Therefore, twin classes are specified to be clique classes in the first and stable set classes in the latter case.
A twin class can also be a single vertex. Even though a single vertex is strictly speaking a stable set as well as a clique, we use the terms clique class and stable set class only for twin classes with at least two elements. We call a twin class with one vertex a singleton class.
In the following, we refer several times to the graph considered in Chapter 3. To the reader’s convenience, we state it here again in Figure 4.1. Note that we slightly varied the equidominating structure. The vertices c1 and c2 form a clique class, the vertices s1
and s2 form a stable set class, and all other vertices are singleton classes.
Now, let T1 and T2 be two twin classes. It is easy to see that either every vertex of T1 is
adjacent to every vertex of T2 or every vertex of T1 is non-adjacent to every vertex of T2.
In the first case we say that T1and T2 see each other and that T1 sees T2 and vice versa.
We also say that a vertex and a twin class see each other, and likewise two vertices. Furthermore, if appropriate, we use expressions for twin classes which are usually used for vertices (for example, a twin class is adjacent to, dominates, is dominated by, et cetera).
For the sake of completeness, we also define the quotient graph Q(G) of a graph G: every twin class of G is a vertex of Q(G) and two vertices are adjacent if and only if the corresponding twin classes see each other. A twin class is a special case of a module and hence the decomposition into twin classes is a special form of a modular decomposition. Therefore, the twin partition can be computed in linear time using one of the modular
As the following observations show, the twin relation is a helpful instrument with regard to minimal dominating sets and thus also with regard to equidomination.
Observation 4.3.3. For every minimal dominating set D and every stable set class S, we have |D ∩ S| ∈ {0, 1, |S|}.
This observation indicates that we have to look carefully at stable set classes with two elements. If a stable set class S contains only two vertices, then Observation 4.3.3 includes all possible cases. In particular, it can not occur that more than one but not all vertices of S are elements of a minimal dominating set. It turns out that stable set classes of size two indeed play a special role (see Lemma 4.3.13 and Definition 4.3.14). Since, in contrast to stable set classes, the vertices of a clique class do dominate each other, the situation is less complicated here.
Observation 4.3.4. For every minimal dominating set D and every clique class C, we have |D ∩ C| ∈ {0, 1}.
The next observation holds not only for stable set classes but also for stable sets in general. It is based on the fact that every maximal stable set is also a minimal dominating set. Therefore, every stable set is contained in at least one minimal dominating set. Observation 4.3.5. For every stable set S and for every equidominating structure (ω, t) it holds that ω(S) ≤ t and hence |S| ≤ t.
Unfortunately, vertices of different twin classes of an equidominating graph can have the same weight in an equidominating structure. This means that the twin partition does not meet the desired condition discussed at the end of the previous section. For example, let us take a look again at the equidominating structure of the graph of Figure 4.1. The vertices c1 and d do not lie in the same twin class since only c1is adjacent to a. However,
they both have the same weight. Therefore, we have to find another way to partition the vertices which needed a careful and intense analysis.
We want to mention that this is one of the significant differences between equidomination and equistability. With respect to an equistable function, two vertices of different twin classes always have different weights [43].