Abstract
An irreversible conversion process is a dynamic process on a graph where a one-way change of state (from state 0 to state 1) is applied on the vertices if they satisfy a conversion rule that is determined at the beginning of the study. The irreversible -threshold conversion process on a graph is an iterative process which begins by choosing a set , and for each step is obtained from by adjoining all vertices that have at least neighbors in . is called the seed set of the -threshold conversion process, and if for some , then is an irreversible -threshold conversion set (IkCS) of . The -threshold conversion number of (denoted by () is the minimum cardinality of all the IkCSs of In this paper, we determine for the circulant graph when is arbitrary; we also find when . We also introduce an upper bound for . Finally, we suggest an upper bound for if and .
1. Introduction
As usual, and denote the numbers of vertices and edges at a graph , respectively. Let be the degree of a vertex ; a graph is -regular if all of its vertices are of degree . The open neighborhood of a vertex is while the closed neighborhood of is . For any undefined term in the paper, we refer to Harary [1]. An irreversible -threshold conversion process on a graph is the process of finding the least number of vertices we need to initially convert in step in order to spread the conversion to all the remaining vertices of the graph according to a conversion rule. This iterative process starts by choosing a seed set , and for each step is obtained from by adjoining all vertices that have at least neighbors in . is called the seed set of the -threshold conversion process, and if for some , then is an irreversible -threshold conversion set (IkCS) of . The -threshold conversion number of (denoted by () is the minimum cardinality of all the IkCSs of . Therefore, and for connected graphs. The first graph model of the irreversible -threshold conversion problem was presented by Dreyer and Roberts in [2] where they determined the value of for paths and cycles For further information on the irreversible -threshold conversion problem on graphs, see [2–6] The circulant graph with the connection set is an undirected graph with the vertex set where two vertices are adjacent if . Therefore, the circulant graph is a cycle, and the circulant graph forms the complete graph . It is obvious that the circulant graph is 4-regular when . Through this paper, we will denote the vertex set by taking into consideration that we exchange the subscript of the vertex by . For further information on the circulant graph, see [7].
Proposition 1 (see [2]).
Proposition 2 (see [2]).
Proposition 3 (see [2]). If is a -regular graph, then is a -conversion set of if and only if is independent.
Note 1: in every figure of this article, we assign the black color to the converted vertices and the white color to unconverted ones
2. Main Results
In this paper, we determine for the circulant graph for arbitrary ; we also find when . We also introduce an upper bound for . Then we suggest an upper bound for if and . Let be a circulant graph on which an irreversible -threshold conversion process is being studied. Since is 4-regular, we define the -unconvertable set of (denoted by ) as follows: which means each vertex of is unconverted and is adjacent to at least vertices of at ; therefore, the conversion cannot reach any vertex of during any step of the process unless at least one of its vertices is converted at . Figure 1 shows a 3-unconvertable set on
Note 2: let be a circulant graph, and let the conversion threshold be We will define a conversion generating path (CGP) as a series of consecutive vertices (a path) on such as (), so that when all of these vertices are converted in a step , they can spread the conversion to the entire graph by converting two new (unconverted) vertices at every following step. The process goes as follows:
the conversion reaches all vertices of the CGP which are
since are adjacent to both then conversion spreads to , and the converted vertices are
since are adjacent to both , then conversion spreads to
The conversion process continues until all vertices of are converted. This goal is achieved on step where represents the number of converted vertices of at .
If is even, then the last two unconverted vertices are converted at the last step, which is
If is odd, then at the next to last step ; two unconverted vertices are converted, and then only one unconverted vertex remains to be converted in the last step which is . Figure 2 illustrates a 5-vertex CGP on .
In this subsection, we determine for arbitrary when .
Theorem 4.
Proof. We know by definition that for any graph , which means that . Let be the seed set of the conversion process. forms a CGP on with , and the process goes as follows:
: we convert
: the conversion spreads to
: the conversion spreads to
The process continues, spreading the conversion to two new (unconverted) vertices each step. If is even, the process ends in step when the last two unconverted vertices () are converted.
If is odd, at the next to last step there are three unconverted vertices left which are (). Two of them () are converted in while the last unconverted vertex () is converted in the last step
We conclude that is an I2CS of which means ; therefore , and we prove the requested.
Theorem 5.
Proof. Since , we need to prove that . Let be the seed set. The process goes as follows:
: we convert
: since the conversion spreads to Similarly, , and we also have which means . Each of the sets , , and forms a CGP on with .
: the conversion spreads to
: the conversion spreads to
The process continues similarly to Theorem 4 until the graph is successfully converted at step if is odd, or at step if is even.
We conclude that is an I2CS of which means ; then , and the requested is proven.
Theorem 6.
Proof. We consider the following cases:
Case 1. . Let be the seed set. The process goes as follows:
: we convert
: since the conversion spreads to Similarly, , and we also have
: the conversion spreads to ; therefore,
: the conversion spreads to ; therefore,
We conclude that .
Case 2. . Let be the seed set. The process goes as follows:
: we convert
: the conversion spreads to which means
: the conversion spreads to which means
We conclude that .
Case 3. . Let be the seed set. The process goes as follows:
: we convert
: the conversion spreads to ; therefore,
: the conversion spreads to which means
: the conversion spreads to ; therefore,
: the conversion spreads to which means
We conclude that .
Case 4. . We start by proving that for . We consider the following subcases:
Case 4.a. ; since , then , and the conversion does not spread after the initial step , which means the process fails. Without loss of generality, the same argument can be applied for any .
Case 4.b. ; since , then at step , we get . However, which means and the spread stops at the end of step . Without loss of generality, this applies to any .
Case 4.c. . In a similar way to the previous two cases, , which means the spread stops at the end of step , and without loss of generality; this applies to any .
Case 4.d. ; then , and the process fails at the end of step . Without loss of generality, this applies to any .
Case 4.e. ; then , but and the spread stops at the end of step . Without loss of generality, this applies to any .
Case 4.f. . Since , then , and the conversion does not spread after the initial step , which means the process fails. Without loss of generality, the same argument can be applied for any .
From subcases 4.a to 4.f, we conclude that
Now, let be the seed set; then at , the conversion spreads to which makes . We notice that each set of four consecutive vertices of forms a CGP on with and . The process goes as follows:
: we convert
1:
The process continues converting two new vertices each step until it ends at if is even, or at if is odd. We conclude that is an I2CS of . Therefore
From (5) and (6), we conclude that for .
From Cases 1–4, we conclude the requested.
Theorem 7.
Proof. Since a CGP is a path of length , and by Proposition 1, therefore, can be an I()CS of . This means Figure 3 illustrates an I2CS of 5 vertices on .
We now consider the following cases for :
Case 1. is odd. Let be a conversion seed set of cardinality . The process goes as follows:
: we convert
:
: . The process stops at the end of step . Without loss of generality, the process applies to all configurations of on the vertices of . Therefore, cannot produce a CGP on , and since we need to convert at least vertices of the path in order to convert it entirely, it is impossible to convert if we initially convert less than vertices at , which means .
Case 2. is even. Let be a conversion seed set of cardinality . The process goes as follows:
: we convert
:
:
The process stops at the end of step . Without loss of generality, the process applies to all configurations of on the vertices of . Therefore, cannot produce a CGP on , and since we need to convert at least vertices of the path in order to convert it entirely, it is impossible to convert if we initially convert less than vertices at , which means .
From Case 1 and Case 2, we conclude that
From (7) and (8), we conclude that .
In this subsection, we determine for . Then, we introduce an upper bound for . Finally, we introduce an upper bound for if and .
Theorem 8.
Proof. We have , and we consider the following cases:
Case 1.
Let be the seed set of the conversion process. It is obvious that is of cardinality . The process goes as follows:
: we convert
: we notice that which means that the conversion spreads to making
: we notice that therefore, the conversion spreads to
The process goes on, and with each step, two new (unconverted) vertices are converted, where and (mod 3). Similarly, and (mod 3). Which means, the next to last step is , and is defined as:
: the conversion spreads to the last remaining vertex (or ) if is odd (or even), respectively, and then the conversion is spread to all vertices of which makes an I3CS of . Therefore
Let be three consecutive unconverted vertices of . At any step of the process, in order to convert any of these three vertices, it needs to be adjacent to three converted vertices. However, this is impossible since each one of them is adjacent to the other two and they are all unconverted, which means is 3-unconvertable if . We imply that any seed set of cardinality on should be distributed as one of the following:
Let us assume that is the seed set. The conversion process goes as follows:
: we convert
with the absence of any unconverted vertex that is adjacent to three vertices of , then which means the spread stops at the end of step . Therefore, the process fails. Without loss of generality, the same result can be obtained if the seed set was or , then:
From (11) and (13), we conclude that if
Case 2.
Similarly to Case 1, a seed set of cardinality is not enough to convert Let the seed set be . It is obvious that . The process goes (similarly to Case 1) as follows:
: we convert
: since the conversion spreads to
: since the conversion spreads to
The process goes on, and with each step, two new (unconverted) vertices are converted, where and (mod 3). Similarly, and (mod 3), which means the next to last step is , and is defined as:
: the conversion spreads to the last two unconverted vertices if , or if . Either way, the process ends successfully at the end of which means is an I3CS of and However, since , we conclude that if
Case 3.
Similarly to Case 1 and Case 2, and in order to avoid getting any version of on , the seed set must take one of the following forms:
Let be the seed set (of cardinality ). The process goes as follows:
: we convert . Whether or , the conversion does not spread to any vertex from because no vertex of is adjacent to three vertices of
: , and the process fails. Without loss of generality, the same result is obtained if the seed set was or ; therefore
Let be seed set of cardinality . The process goes as follows:
: we convert
: the conversion spreads to
: the conversion spreads to
Similarly to Case 1 and Case 2, by the end of each step, two new (unconverted) vertices are converted, and the process continues until the last step when the last two unconverted vertices which are (in case is even), and (in case is odd) get converted and then the conversion process reaches the entire graph. We conclude that:
From (16) and (17), we conclude that
From all the previous cases, we conclude the requested.
Proposition 9. A set of consecutive unconverted vertices in is 3-unconvertable.
Proof. Let there be a conversion process on , and at the initial step , let the set be a set of unconverted vertices. Every vertex of is of degree 4 and is adjacent to two other vertices of . Since , it is impossible for any of these vertices to satisfy the conversion condition at any step of the conversion process even if all the vertices of get converted, which means is 3-unconvertble.
Theorem 10. For we have:
Proof.
Let be the seed set of the conversion process. We implied in Proposition 9 that the process fails if contains four (or more) consecutive vertices, which means contains at least vertices. Let be the following sets:
In order to avoid having four consecutive unconverted vertices, must contain either . We assume that ; we notice that the vertices of divide into subgraph, each of which consists of four consecutive vertices, and only one of these vertices is converted at . We denote these subgraphs by , when:
We take into consideration that if , respectively. The last vertices form a minisubgraph of 1,2, and 3 vertices if , respectively. We imply that for , the conversion does not spread to any vertex of because none of them is adjacent to three vertices of which means , and the process fails. This means Let be a set of five consecutive vertices where the middle vertex is the only converted vertex of ; then the four unconverted vertices of form a 3-unconvertable set on and that is because each of them is adjacent to two other unconverted vertices of , which means there cannot be a step when any of these four vertices satisfies the conversion rule (being adjacent to three or more converted vertices). We conclude that there cannot be two consecutive subgraphs , of the subgraphs identified previously, or else, a version of will be created consisting of the last two vertices of and the first three vertices of ; this means that we need to add at least one converted vertex to one of every two consecutive subgraphs. Therefore
We consider the following cases for :
Case 1. . We consider two subcases:
Case 1.a. .
In this subcase, we have an even number of subgraphs on and . Let be the following set: . Let be the seed set. It is obvious that . The conversion process goes as follows:
: we convert
: the conversion spreads to
: the conversion spreads to
By the end of step , the conversion reaches all vertices of ; therefore, is an I3CS of and , and from (21), we conclude that when .
Case 1.b. and .
This subcase is similar to subcase 1.a with the only difference of having an odd number of subgraphs. This means we need to convert one additional vertex . The conversion process goes as follows:
: we convert
: the conversion spreads to
: the conversion spreads to the remaining vertices which are
Therefore, is an I3CS of and since , according to (21), we conclude that if and . Figure 4 illustrates an I3CS of 8 vertices on .
Case 2. . We consider two subcases:
Case 2.a. .
In this subcase, we have an even number of subgraphs on and . Let and be the same sets identified in subcase 1.a. Let the seed set be . In a similar process to the one in subcase 1.a, all vertices of by the end of step . However, the five consecutive vertices form a version of which was identified in Theorem 8 as containing a 3-unconvertable set. In addition to that, since taking out any vertex from results in a version of either on , we conclude that in this subcase. Let be the seed set of cardinality ; the process goes as follows:
: we convert
: the conversion spreads to
: the conversion spreads to which means is converted at this step as well
: the conversion spreads to
By the end of step , the conversion reaches all vertices of ; therefore, is an I3CS of , and since , we conclude that if .
Case 2.b. and .
This subcase is similar to subcase 2.a with the only difference of having an odd number of SGs. This means that similarly to subcase 1.b, we need to convert one additional vertex from this last subgraph ; the process goes as follows:
: we convert
: the conversion spreads to
: the conversion spreads to
: the conversion spreads to
: the conversion spreads to
By the end of step , the conversion reaches all vertices of . Therefore, is an I3CS of , and since , this means if and .
Case 3. . We consider two subcases:
Case 3.a. .
In this subcase, we have an even number of subgraphs on and . Let and be the same sets identified in subcase 1.a. Let the seed set be . In a similar process to the one in subcase 1.a, all vertices of by the end of step . However, the five consecutive vertices form a version of , and since taking out any vertex from results in a version of either , we conclude that in this subcase as well. Let be the seed set of cardinality ; the process goes as follows:
: we convert
: the conversion spreads to which means are converted in this step
: the conversion spreads to which means is converted at this step
By the end of step , the conversion reaches all vertices of ; therefore, is an I3CS of , and since , we conclude that if .
Case 3.b. and .
By following the same argument in subcase 1.b and subcase 3.a, let be the seed set of cardinality ; the process goes as follows:
: we convert
: the conversion spreads to the vertices which means are converted in this step
: the conversion spreads to the vertices
: the conversion spreads to the last unconverted vertex , and the entire graph is converted
We conclude that if and .
Case 4. .
We consider two subcases:
Case 4.a. .
In this subcase, we have an even number of subgraphs on and . Let and be the same sets identified in subcase 1.a. Let the seed set be . In a similar process to the one in subcase 1.a, all vertices of by the end of step . However, the five consecutive vertices form a version of , and since taking out any vertex from results in a version of either , we conclude that in this subcase as well. Let be the seed set of cardinality ; the process goes as follows:
: we convert
: vertices are converted, which means is converted at this step
: the conversion spreads to which means is converted at this step
: are converted
: are converted
By the end of step , the conversion reaches all vertices of ; therefore, is an I3CS of , and since , we conclude that if .
Case 4.b. and . We consider two subcases:
Case 4.b..
This proof is equivalent to proving that It is obvious by definition that . Let be a seed set of cardinality 3 and defined as . The process goes as:
which means ; therefore,
Case 4.b..
Let ; then that would make the following nine vertices unconverted which means creating several versions of , and the process fails. Now let be the seed set. We consider the following options for (i)If , then are four consecutive unconverted vertices; therefore, they form a version of , and the process fails. However, we notice that(ii)If , then () form a version of , and since is 3-unconvertable, then the process fails(iii)If , then form a version of , () form a version of , and the process failsWe conclude cannot be a I3CS of when and which means that if and .
Now let be the seed set of cardinality ; the process goes as follows:
: we convert
: the conversion spreads to
: the conversion spreads to
the last two unconverted vertices get converted which means the entire graph is converted at the end of this step
We conclude that is an I3CS, and therefore, if and .
From all cases and subcases, we conclude the requested.
Theorem 11. For
Proof. We implied in Proposition 9 that the conversion process fails if there are five consecutive unconverted vertices on at , which means the conversion seed set cannot consist of less than vertices. Let ; we assume that , and we notice that the vertices of divide the first vertices of into subgraphs, each of which consists of four consecutive unconverted vertices followed by one converted vertex. We denote them by . The two adjacent subgraphs together have two converted vertices {} and eight unconverted vertices which are . We notice that the set consists of four unconverted vertices each of which is adjacent to two vertices of which means is 3-unconvertable. Therefore, the process fails if , Without loss of generality, the same argument applies to any . Let us now try to find a configuration of converted vertices that prevents having any unconvertable sets and at the same time guarantees total conversion of . We imply that converting and applying this configuration to the neighboring subgraphs achieves the requested for as shown in Figure 5.
Therefore, we apply this configuration to every two adjacent subgraphs. As for the remaining , we will need to convert additional vertices in order to convert them. In that regard, we consider the following cases of :
Case 1.
Let be the seed set. The process goes as follows:
: we convert
: the conversion spreads to
: the remaining unconverted vertices which are get converted
end of step , the entire graph’s vertex set is converted. We conclude that is an I3CS of cardinality , which means if . Figure 6 illustrates that .
Case 2.
Let be the seed set. The process goes as follows:
Therefore, is I3CS of which means if .
Case 3.
Let be the seed set. The process goes as follows:
Therefore, is I3CS of which means if .
Case 4.
Let be the seed set. The process goes as follows:
Therefore, if .
Case 5.
We consider the following subcases:
Case 5.a. .
Let the seed set be which is of cardinality . Then
Therefore, .
Case 5.b. and
Let the seed set be . The process goes as follows:
Therefore, if and .
Case 6. .
Let be the seed set. The process goes as follows:
which means if .
Case 7.
We consider the following subcases:
Case 7.a. .
Let the seed set be which is of cardinality . The process involves the following steps:
Therefore, .
Case 7.b. and
Let be the seed set. The process goes as follows:
We conclude that if and .
Case 8. . We consider the following subcases:
Case 8.a. .
Let the seed set be which is of cardinality . The process involves the following steps:
Therefore, .
Case 8.b. and
Let be the seed set. The process goes as follows:
Therefore, if and
Case 9. . Let be the seed set. The process goes as follows: Therefore, if . Figure 7 illustrates that .
Case 10. . We consider the following subcases:
Case 10.a. .
Let the seed set be which is of cardinality . The process involves the following steps:
Therefore, .
Case 10.b. .
Let the seed set be which is of cardinality . The process goes as follows:
Therefore, .
Case 10.c. and
Let be the seed set. The process goes as follows:
Therefore, if and
From all the previous cases and subcases, we conclude the requested.
Theorem 12. For and :.
Proof. Proposition 9 implies that the conversion process fails if there are consecutive unconverted vertices on at . We divide the vertices of into subgraphs denoted by . Now we try to find a configuration of converted vertices of a random subgraph () at so that when applied to all the subgraphs, it results in converting all of We consider the following cases for :
Case 1. is even. Let the configuration of converted vertices we apply to at be . This means we convert vertices from each subgraph. As shown in Figure 8, in step , the conversion spreads to . In the following step , the conversion spreads to . In step , the conversion spreads to . In step , the configuration converts entirely.
Without loss of generality, by applying the same configuration to all subgraphs, we form an I3CS of cardinality . We denote it by , and the process goes as follows:
Therefore, is an I3CS which means and is even.
Case 2. is odd. Let the seed set be . The process goes as follows: Therefore, is an I3CS which means and is odd. Figure 9 illustrates how converting at results in converting entirely at the end of step , taking into consideration that and .
Without loss of generality, the same argument applies to all subgraphs. From Case 1 and Case 2, we conclude the requested.
Data Availability
No data was used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Faculty of Science, Tishreen University, Syria.