Convex and weakly convex domination in prism graphs
Abstract.
For a given graph and permutation the prism of is defined as follows: , where is a copy of , and , where and denotes the copy of in .
We study and compare the properties of convex and weakly convex dominating sets in prism graphs. In particular, we characterize prism fixers and doublers. We also show that the differences and can be arbitrarily large, and that the convex domination number of cannot be bounded in terms of
1. Introduction
Let be an undirected graph, let be a permutation of its vertex set and let be a vertexdisjoint copy of . We denote the copy of a vertex in by . If is a set of vertices of , then denotes the copy of in , i.e. the set By and we denote the vertex sets of and , respectively. The prism graph is a graph with the vertex set and the edge set where
We say that a set dominates if We denote this by A set is called a dominating set of the graph if it dominates i.e. if The minimum cardinality of a dominating set in is called the domination number of and denoted A minimum dominating set of a graph is sometimes called a set.
Prism graphs were first defined in [2] and the problem of domination in prism graphs was first studied by Burger, Mynhardt and Weakley [1].
In a connected graph a set of vertices is said to be convex if for every pair of vertices the set contains all vertices of every shortest – path. A convex dominating set of is a dominating set of which is convex. The minimum cardinality of a convex dominating set of is called the convex domination number of and denoted as . A set of is a convex dominating set of cardinality .
A set of vertices in a connected graph is said to be weakly convex if for every pair of vertices it contains all vertices of at least one shortest –. A weakly convex dominating set is a dominating set of which is weakly convex. The minimum cardinality of a weakly convex dominating set of is called the weakly convex domination number and denoted as . A set of is a minimum weakly convex dominating set.
A set is said to be connected if it induces a connected subgraph, i.e., if for every pair of vertices there exists a – path contained entirely in . A connected dominating set of is a dominating set which is connected. The connected domination number of , denoted as , is the minimum cardinality of a connected dominating set of . A set of is a connected dominating set of cardinality
In this paper we compare the properties of convex and weakly convex sets in prism graphs, particularly convex and weakly convex dominating sets. We also generalize some known properties of convex domination in prism graphs to weakly convex and connected domination.
2. Connected, convex and weakly convex domination
It is clear that the notions of convex, weakly convex and connected domination are closely related. Since every convex set is weakly convex and every weakly convex set is connected, it is easy to see that . Many, but not all, properties of connected sets can be extended to both convex and weakly convex sets.
Lemma 1.
If is a connected graph with and is a permutation of , then
Proof.
Obviously, if , then
If and is a set of , we denote and Since the only vertices in dominated by are in it follows that and therefore Thus, Since it follows that and therefore is not connected. Thus, every connected dominating set of has cardinality at least ∎
Since Lemma 1 also applies to convex and weakly convex domination, that is, for any graph and permutation of we have and
Observation 2.
For any connected graph and permutation of let be a dominating set of such that:

,

is connected,

is connected,

Then is a connected dominating set of size in .
Proof.
Since and , it is clear that that is, is a dominating set in . The sets and are connected and it follows that is connected. Hence is a connected dominating set of ∎
Note that this particular result cannot be extended to convex domination as, for example, if then for any set , where and are not empty, the set is not convex. For connected domination, however, we can use it to show exactly when the bound from Lemma 1 is achieved.
Proposition 3.
Let be a connected graph with and let be a permutation of Then if and only if there exists a set of and a vertex such that:


is connected,


is connected.
Proof.
Let be a set of vertices in It follows from Observation 2 that is a connected dominating set of size in , so . By Lemma 1, . Thus,
Now let be a connected dominating set of size in and let and
Since dominates it is necessary that and
Now let and . If then , a contradiction. Thus, is a set, and
Since is connected, there exists a vertex such that
Suppose Then is a dominating set and This implies
Finally, if is not connected, then there exists a vertex in each of its components such that , as otherwise the set would not be connected. Then . Similarly, if is not connected, then . ∎
Again, the above is not true for convex domination, however, in Section 4 we prove a related property of weakly convex dominating sets.
3. Generalizing some properties of convex domination
Convex domination in prism graphs was studied by Lemańska and Zuazua in [3], where they prove the following theorem.
Theorem 4 ([3]).
Let be a connected graph. If
and then for every permutation of .
The proof of Theorem 4 relies some properties of convex dominating sets, which they prove in the same paper. We will now compare these properties with those of weakly convex dominating sets.
Theorem 5 ([3]).
For any connected graph :

If then and are both convex dominating sets of for any permutation

If then there exist permutations and such that is not a convex dominating set of and is not a convex dominating set of
It follows that if then for any . The weakly convex domination number has a similar property.
Theorem 6.
Every connected graph has the following properties:

If then and are weakly convex dominating sets of for every permutation

If , then there exist permutations and such that is not a weakly convex dominating set of and is not a weakly convex dominating set of
Proof.
Obviously, for any permutation and are dominating sets of For any pair of vertices the shortest – path containing at least one vertex from has length at least Thus, if then and contains at least one shortest – path in Similarly, if then must contain a shortest – path in for every
Now, let be at least and let be a pair of vertices such that If is a permutation such that and are adjacent, then . Thus, is not a weakly convex dominating set in Similarly, for , the set is not weakly convex. ∎
The first part of Theorem 6 implies that for any graph with diameter at most
Proposition 7 ([3]).
For a connected graph and permutation of , let be a convex dominating set of and let and Then has the following properties:

If , then and

If and then there exists at least one such that .
The above can be generalized to weakly convex and connected domination.
Proposition 8.
For a connected graph and permutation of , let be a connected dominating set of and let and Then has the following properties:

If , then and

If and then there exists at least one such that .
Proof.
If then . But every vertex in only dominates one vertex in It follows that if is a dominating, then implies . The same reasoning applies if Thus, if then
If both sets and are nonempty, then contains a pair of vertices Since the set is connected, there exists a  path contained entirely in . Each vertex has only one neighbor in namely Hence, every path connecting with contains a pair of vertices and This shows that contains a vertex such that ∎
Since every weakly convex dominating set is a connected dominating set, we also have the following.
Corollary 9.
For a connected graph and permutation of , let be a connected dominating set of and let and Then has the following properties:

If , then and

If and then there exists at least one such that .
Lemma 10 ([3]).
Let be a connected graph in which . Let be a convex dominating set of and let and Then the set has the following properties:

If , then is a convex dominating set of

If , then is a convex dominating set of
Again, we can prove a similar property for weakly convex domination.
Lemma 11.
Let be a connected graph in which . Let be a weakly convex dominating set of and let and Then the set has the following properties:

If , then is a weakly convex dominating set of

If , then is a weakly convex dominating set of
Proof.
If , then clearly dominates as does not dominate any part of It follows that is a dominating set of . For any two vertices the shortest possible path in containing a vertex from has length at least . If then any shortest path must be contained in . Thus, for every the set contains a shortest path. Hence, is a weakly convex dominating set of .
Similarly, if , then is a dominating set of because does not dominate any part of . is also convex, because for any no shortest path passes through ∎
Note that, unlike Theorem 6, the above is does not hold for every such that For example, if and the set is a weakly convex dominating set of . The set contains and it is not a weakly convex set.
Since so many of the results in [3] can be generalized to weakly convex domination, one could expect a weakly convex analogue of Theorem 4 to be true as well. However, Lemańska and Zuazua’s proof relies on showing that if is a conncected graph with , then a dominating set of with less than vertices cannot be convex. This is based on properties of convex sets which weakly convex sets do not have. Thus there appears to be no reason why such a set cannot be weakly convex and therefore the following conjecture seems more likely.
Conjecture 1.
There exists a connected graph with and permutation of such that and
4. Convex and weakly convex domination in
We now consider the special case where the permutation A graph is called a prism fixer if and a prism doubler if A graph is called a universal fixer if for every permutation of and a universal doubler if for every Prism fixers are characterized in [4] and universal fixers in [5] and [6]. Prism doublers and universal doublers are studied in [1].
Similarly, a graph such that is called a prism fixer and a graph with is called a prism doubler. A universal fixer is a graph such that for every and a universal doubler is a graph such that for every
We begin this section by studying some properties of convex and weakly convex sets in
Lemma 12.
If is a weakly convex set in , then and are weakly convex sets in
Proof.
Let and be any two vertices in . Since the set is weakly convex, it contains a shortest – path in . It is easy to see that . Thus also contains a shortest – path in . Since the same applies to the set is also weakly convex.
Now let be a shortest – path in and let . The path is a shortest – path in contained in . Thus is a weakly convex set in ∎
The same applies to convex sets.
Lemma 13.
If is a convex set in , then and are convex sets in
Proof.
Let and be any two vertices in . Since the set is convex, it contains every – path of length in . The shortest – path in containing a vertex from has length Thus, and contains every shortest – path in Since the same applies to the set is also convex.
For two vertices we have Every shortest – path in has the form for some shortest – path in . Since contains that – path, is a convex set in ∎
Weakly convex sets also have the following property.
Lemma 14.
A set is weakly convex if and only if and:

, and are weakly convex sets in

For every , the shortest  path in contains a vertex from
Proof.
Let be a weakly convex set in and let Since there is no shortest  path in containing a vertex from , the set must contain a shortest  path, which is also a shortest  path in . The same applies to Thus and are weakly convex sets in . Now let and If is a  path in shorter than the shortest  path in then is a  path in shorter than the shortest  path in and is not weakly convex. Thus, is a weakly convex set in . Furthermore, since and it is clear that
Now let be a set satisfying conditions (1)(2). By Lemma 12 and are convex in Let The set contains a shortest  path where Since and are weakly convex, the paths and are contained in and , respectively. Thus is a shortest  path in and contains all of its vertices. ∎
This is not the case with convex sets. If a convex set in contains a pair of vertices it must also contain and This leads to some differences between convex and weakly convex domination.
Theorem 15.
If is any connected graph, then
Proof.
Let be a set of . Obviously, dominates . By Lemma 13, is a convex set in . Hence, is clearly a convex dominating set of . The set is also a convex dominating set of
Let be a set in If then obviously Similarly, if then . If contains vertices and it must also contain every shortest  path in . Thus, contains and all shortest  and  paths. Therefore, in this case, for some and is a set of . ∎
As a result, we have the following.
Corollary 16.
Every connected graph has the following properties:

is a prism fixer if and only if

is a prism doubler if and only if
Proof.
If then, obviously, . Thus, by Theorem 15, we have
The set is a convex dominating set of Thus, if then, by Theorem 15, is a minimum convex dominating set of .
By Theorem 15, if and only if Thus, if and only if . ∎
Since every universal fixer is a prism fixer, and every universal doubler is a prism doubler, we also have the following corollary.
Corollary 17.
Let be a connected graph. Then:

If is a universal fixer, then

If is a universal doubler, then
Weakly convex domination has a somewhat similar property.
Theorem 18.
If is a connected graph, then .
Proof.
Obviously, is a dominating set in By Lemma 12, it is also a weakly convex set in Thus,
If is a set of then is a dominating set in as and Since Lemma 12 implies that is a (not necessarily minimal) weakly convex dominating set in and thus . ∎
However, thanks to Lemma 14, is not necessarily equal to . In fact, the following is true.
Theorem 19.
Let be a connected graph. The graph has a weakly convex dominating set of cardinality if and only if has a weakly convex dominating set which can be partitioned into three nonempty sets such that and:

and are weakly convex,

For every , the shortest – path in contains a vertex from ,

and .
In particular, if and only if has a set such that conditions (1)(3) are fulfilled and .
Proof.
Let be a weakly convex dominating set of a connected graph . The set is a dominating set of as and By Lemma 14, is a weakly convex set. Thus, has a weakly convex dominating set of size
Now let be a weakly convex set of and let and . If either or was an empty set, then would be either or By Lemma 14 the sets and are convex and for every there is a shortest  path in containing a vertex from . Since the only veritces in dominated by are in , it is clear that Similarly, , as the only vertices in dominated by are in Thus is a weakly convex domiating set of such that, for and conditions  are fulfilled. ∎
For example graph in Fig. 1 has such a set. As a result .
5. Upper and lower bounds
It is well known that the inequalities
(1) 
hold for any graph and any permutation of its vertex set. At the conference Colorings Inependence and Domination in 2015 Rita Zuazua conjectured that similar inequalities hold for convex and weakly convex domination, i.e.
(2) 
and
(3) 
However, this is not true in general. The smallest counterexample is the path with and the permutation . In this case while .
For a star with and the permutation , where is the central vertex and is one of the other vertices, we have and while . Thus, the upper bounds in (2) and (3) not hold for
Furthermore, for every there is a graph and permutation such that .
Let us begin with the cycle and the permutation The weakly convex domination number of is , but the graph can be dominated by a weakly convex set with only vertices: .
In fact, the difference can be arbitrarily large. For any we can construct a graph as follows (see Fig. 2).

Take copies of . Denote the th copy of the vertex by

Replace the vertices with a single vertex
The permutation is defined as Then and