Computing the hull number in toll convexity

04/30/2019 ∙ by Mitre C. Dourado, et al. ∙ UFRJ 0

A walk W between vertices u and v of a graph G is called a tolled walk between u and v if u, as well as v, has exactly one neighbour in W. A set S ⊆ V(G) is toll convex if the vertices contained in any tolled walk between two vertices of S are contained in S. The toll convex hull of S is the minimum toll convex set containing S. The toll hull number of G is the minimum cardinality of a set S such that the toll convex hull of S is V(G). The main contribution of this work is an algorithm for computing the toll hull number of a general graph in polynomial time.

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1 Introduction

We consider finite, simple, and undirected graphs. For a graph , its vertex and edge sets are denoted by and , respectively, while the open and the closed neighborhoods of a vertex are denoted by and , respectively. Recall that a walk between vertices and of a graph is a sequence of vertices such that , for , , and . As a motivation, consider that a graph models a space containing two points with huge gravitational force, represented by vertices . Thus, a valid trajectory of a spacecraft launched from with destination to is represented by a walk that contains exactly two vertices of and of , since the necessary energy for to move away from is all wasted in the take off and once reaches the neighborhood of , it is imediatilly absorbed by . This scenarium prevents from passing through the neighborhood of a second time, because in this case would be absorbed by and the mission will be failed. Path convexities has gained attention in the last decades [11, 14, 16, 19], and this kind of relaxation of path originated the toll convexity [2, 15]. A tolled walk between and , or a tolled -walk, is a walk in which , , and if , then is the only neighbor of and is the only neighbor of in .

A family of subsets of a finite set is a convexity on if and is closed under intersections [20]. Given a graph , a set is toll convex if the vertices contained in any tolled walk between two vertices of are contained in ; and is toll concave if is toll convex. The toll interval of is belongs to some tolled -walk. The toll interval of is if and otherwise. If , then is said to be a toll interval set of and the minimum cardinality of a toll interval set of is the toll number of . The toll convex hull of , denoted by , is the minimum toll convex set containing . If , then is said to be a toll hull set of and the minimum cardinality of a toll hull set of is the toll hull number of . Note that if is an induced subgraph of and is a tolled -walk of , then is also a tolled -walk in . Hence, and . For shortness, we will drop the superscript and subscript indicating the graph and the convexity when there is no ambiguity.

For , denote by the graph obtained by the deletion of the vertices of ; and by the subgraph of induced by . If every two vertices of are adjacent, then is a clique of . Vertices are twins if . Vertex  is simplicial in if is a clique. If is a clique, then is said to be a complete graph. The neighborhood of is and the border of is and . We will also use and, for a family os sets , will stand for . A vertex of a toll convex set is extreme in if is also a toll convex set. Denote the set of toll extreme vertices of by . It is clear that is subset of every toll interval set and of every toll hull set of and the every toll extreme vertex is a simplicial vertex but the converse is not always true.

In the well-known geodetic convexity [13, 19], monophonic convexity [11, 12], and convexity [10, 16] all above concepts are analogously defined by replacing “tolled walk” by “shortest path”, “minimal path”, and “path of order three”, respectively. In the geodetic convexity, determining whether the hull number is at most is -hard for general graphs [7], -complete for partial cube graphs [1] and chordal graphs [5], and solvable in polynomial time for unit interval graphs, cographs, split graphs [8], cactus graphs, -sparse graphs [3], distance hereditary graphs [17], (,triangle)-free graphs [4]. In the convexity, this problem is -hard even for bipartite graphs with maximum degree  [7], and can be solved in polynomial time for block graphs and chordal graphs [6]. However, the monophonic hull number can be computed in polynomial time for general graphs [9]. In the toll convexity, it is known that the hull number of every tree different of a caterpillar is equal to 2 [2].

A graph is an interval graph if every vertex of can be associated with an interval of a straight line such that two vertices of are neighbors if and only if the corresponding intervals intersect. Given a convexity on the vertex set of , we say that is a convex geometry under if every -convex set of is equal to the -convex hull of its -extreme vertices. In [2], it was shown that the interval graphs are precisely the graphs which are convex geometries in the toll convexity. They also characterized the toll convex sets of a general graph and of some graph products. In [15], the toll number of the Cartesian and the lexicographic product of graphs are studied, where some characterizations are presented.

The text is organized as follows. In the next sextion, we present the notion of hull representing family, which plays an important role in the proposed algorithm and can be an useful tool for further works dealing with the hull number. In Section 3, we present a polynomial-time algorithm for computing the toll hull number of a general graph. In the conclusions, we discuss that this result leads to an algorithm for generating all minimum toll hull sets of a general graph with polynomial delay and to a characterization of the toll extreme vertices of a graph.

2 Hull characteristic family

We begin this section proving useful properties of tolled walks.

Lemma 2.1.

Let be a graph, let , let such that is connected, and let . The following sentences are equivalent.

  1. There is a tolled -walk containing vertices of ;

  2. There is a tolled -walk containing vertices such that , , , and ;

  3. .

Proof.

Let be a tolled -walk containing vertex . Since separates from , contains at least two occurrences and of vertices in such that appears between and in . By definition, and . Furthermore, we can write such that . Now, the assumption that is connected guarantees that there is a -walk containing all vertices of . Since and , the walks and can be combined to form a tolled -walk containing all vertices of as desired.


is direct from definition. ∎

The following intereseting consequence of Lemma 2.1 does not work in general for other path convexities.

Corollary 2.2.

If induces a connected graph and is toll concave, then any set that induces a connected graph and contains is toll concave.

Before introducing the hull characteristic families, we recall an useful result.

Lemma 2.3.

[2] A vertex is in some tolled walk between two non-adjacent vertices and if and only if does not separate from and does not separate from .

Observe that Lemma 2.3 can be used to test whether a vertex is toll extreme, a set is toll concave, and to show that is a clique for every toll concave set .

If is a concave set of a convexity on a set , then every hull set of has at least one vertex of . We define the granularity of under as the maximum integer such that every hull set of has at least vertices of . Let be a family of pairwise disjoint concave sets of . The granularity of is the sum of the granularities of its members. We say that is a hull characteristic family of if the hull number of is equal to the granularity of .

The problem of computing the hull number of can be reduced to the one of finding a hull characteristic family of and computing the granularity of each of its members. The family formed only by is itself a trivial hull characteristic family of , but it brings no advantage of the use of this notion for determining the hull number of . The number of hull characteristic families of can be an exponential on the cardinality of . For instance, every partition of the vertex set , where is a complete graph, is a hull characteristic family of the toll convexity of , since the toll hull number of is if is a complete graph. An example of a non-trivial hull characteristic family in toll convexity is the family of vertices of the graph of Figure 1. One can use Lemma 2.3 to see that the members of are really toll concave sets. In fact, this lemma can be used to show that all vertices of are extreme vertices, then . Since is not a toll hull set of , the toll hull number of is at least 4. Now, one can use Lemma 2.3 again to prove that is a toll hull set of concluding that and also that the toll hull number of is 4.

Figure 1: Graph .

3 The algorithm

The central idea of the proposed algorithm is to find a toll hull characteristic family of the input graph such that the granularity of each member of can be determined in polynomial time. In order to get this, initially, one family of sets is constructed such that, during the algorithm, its member, that are not toll concave, are getting bigger, possibly concatenating with other members of so that, at the end, the toll concave sets of form the desired family. The following classification of the toll concave sets of a graph is useful to accomplish this task.

Lemma 3.1.

If is a toll concave set of a graph , then if the type of is and if the type of is .

Proof.

Let be a toll concave set of with type . The case is trivial. For the case , suppose for contradiction that is a toll hull set of such that . Since is toll concave and is not, for some , there is a tolled -walk containing some vertex . However, since because , separates from , which contradicts Lemma 2.3.

Finally consider . We claim that all vertices of are extreme vertices. Suppose the contrary and let be a tolled -walk containing some vertex . Since is toll concave, at least one, say , belongs to . But and are twins, because is a clique and every vertex of is universal to by definition of type 3. This contradicts Lemma 2.3 because separates from . ∎

An example of a toll concave set with granularity strictly bigger than its type is the set of Figure 1, since the type of is and because vertices are toll extreme vertices of the graph.

We need some aditional definitions. Consider a graph . We say that separates vertices if there is a -path in but there is no one in ; that is a separator of if separates some pair of vertices of ; and that is a clique separator of if is a clique and a separator of . We say that is reducible if it contains a clique separator, otherwise it is prime. A maximal prime subgraph of , or mp-subgraph of , is a maximal induced subgraph of that is prime. An mp-subgraph of a reducible graph is called extremal if there is an mp-subgraph different of such that, for every mp-subgraph different of , it holds . As an example, consider the graph of Figure 1. The mp-subgraphs of are induced by the following sets , , and . The following result states an useful property of reducible graphs.

Lemma 3.2.

[18] Every reducible graph has at least two extremal mp-subgraphs.

Lemma 3.3.

If is a non-extremal mp-subgraph of , then is disconnected.

Proof.

Let be a non-extremal mp-subgraph of and let and be mp-subgraphs of such that there is no mp-subgraph of different of containing . Then, there are vertices and . There are also and such that separates from and separates from . Therefore separates from . ∎

The following result on the monophonic convexity solves the problem when the input graph is prime.

Theorem 3.4.

[9] If is a prime graph that is not a complete graph, then every pair of non-adjacent vertices is a monophonic hull set of .

Corollary 3.5.

Let be a prime graph. If is a clique, then ; otherwise every two non-adjacent vertices form a toll hull set of .

Proof.

If is a complete graph, it is clear that is the only toll hull set of . If is a not a complete graph, the result follows from Theorem 3.4 because for any set . ∎

Once a toll concave set is found by the algorithm, it is added to and keep this way until the end of the algorithm. Therefore, it will be a member of the toll hull characteristic family constructed for the input graph. Therefore, one can determine its type and choose the vertices of that compose the minimum toll hull set that will be returned. The possible selections appear as numbererd choices in the algorithm and are detailed in the sequel.

input : A graph
output : A minimum toll hull set of
1 if  is a clique then
2        return
3if  is prime then
4        return two non-adjacent vertices of
5compute the mp-subgraphs of is the vertex set of a non-extremal mp-subgraph of is the vertex set of an extremal mp-subgraph of for  do
6        if  is toll concave of type  then
7              apply Choice 1
8       if  is toll concave of type  then
9              apply Choice 4
10       if  is toll concave of type  then
11              add to
12       
13 while there is that is not toll concave and for some  do
14        and and if  is toll concave of type  then
15               let be the number of members of such that is toll concave if  and  then
16                     if possible, apply Choice 2; else apply Choice 3
17              if  and  then
18                     if possible, apply Choice 5; else apply Choice 6
19              if  and  then
20                     if possible, apply Choice 7; else apply Choice 8
21              
22       
is a toll concave set of return
Algorithm 1 Minimum toll hull set
Choice 1.

add to such that and has a non-neighbor in .

Choice 2.

add to for which there are with such that , , there is , and there is .

Choice 3.

add to for which there are with such that , , and there is .

Choice 4.

add and to such that and are non-adjacent vertices of .

Choice 5.

add and to such that there is with and both, and , have non-neighbors in .

Choice 6.

for , add to such that there is with , has a non-neighbor in , and .

Choice 7.

add to such that there is with and has a non-neighbor in .

Choice 8.

add to such that there is with .

Lemma 3.6.

At any moment of Algorithm 

1, the family satisfies the following sentences.

  1. if , then is non-empty and is connected;

  2. the members of are pairwise disjoint;

  3. if and is not a clique, then is disconnected.

Proof.

After line 1, each member of is a different extremal mp-subgraph of . Then itens (1) and (2) hold at this moment. After line 1 of each iteration of the While loop, one member is added to which is the union of some members removed from plus some members of , which are mp-subgraphs of do not belonging to any other member of . It is clear that this operation preserves the property that the members of form a partition of a subfamily of the mp-subgraphs of each one containing at least one extremal mp-subgraph and that is a connected graph.

Since an extremal mp-subgraph contains a vertex not belonging to any other mp-subgraph, item  holds and the fact that the intersection between two mp-subgraphs and is a subset of implies item .

For item , let be such that is not a clique and let be two non-adjacent vertices. Recall that is the union of some mp-subgraphs of and that, if is an mp-subgraph of containing , then there is an mp-subgraph of not contained in such that . Analogously, there are and for . Note that can be equal to , but and . Now, observe that separates from but does not separate from . Since , it follows that and belong to different connected components of . ∎

The following result guarantees that if is a toll concave set constructed in Algorithm 1 by the union of other sets, then at most two of these sets are such that is toll concave. Furthermore, the type of is 1.

Lemma 3.7.

Let and be obtained in lines 1 and 1 of the same iteration of the While loop of Algorithm 1, respectively. If is toll concave, then has at most two toll concave sets and each of them has type .

Proof.

First, suppose for contradiction that has three toll concave sets and . Since is toll concave, and the border of every toll concave set is a clique, we conclude that is a clique. Hence, every pair for which there is a tolled -walk containing some vertex of satisfies .

By Lemma 3.6, the sets and are pairwise disjoint. This implies that for at least one of them, say , it holds . Furthermore, we have that for , because each of and has at least one non-neighbour in , for , and is a clique. Observe that has at least two occurrences and of vertices of such that and . Now, since , every vertex of has at least one neighbor in , and is connected by Lemma 3.6, it holds, by Lemma 2.1(2), that , which contradicts the assumption that is toll concave. Therefore, has at most toll concave sets.

Now, suppose for contradiction that is a toll concave set of type 2 or 3. This means that every vertex of is universal to . Therefore, we have that because each of and has at least one non-neighbor in , , and is a clique. As in the previous case, this implies that , which means that is not a toll concave set, a contradiction. ∎

Now, we show that all choices done by the algorithm are possible in the specific situations that they are done.

Lemma 3.8.

Consider Algorithm 1. Choices  and are always possible in lines  and 1, respectively. Choice or is always possible in line 1.

Proof.

Choice 1 is always possible for a toll concave set having type 1 and Choices 4 and 8 are always possible for a toll concave set having type 2.

Then consider line 1. By definition of type 1, there is with a non-neighbor in . If , it is clear that for some . Then, set , and set any other member of as . If , then there is only one member of containing . Set such member as . Since, for any member , it holds that and are disjoint by Lemma 3.6, any member of can be chosen as and any member of as .

Finally, consider line 1. Since is not toll concave and is a clique, there are vertices for which there is a tolled -walk containing vertices of . Since is toll concave of type 2, every vertex of is universal to . Therefore, and we can write and , for , which matches with Choice 5 or 6. ∎

The next result is essential to show that only one vertex suffices for every set belonging to the toll hull characteristic family constructed by the algorithm such that has type 1. We need one more definition. If is such that there is a maximal set containing that induces a connected graph, then we denote by the vertex set of the connected component of containing .

Lemma 3.9.

If of Algorithm 1 is such that has type , then and, for every , it holds .

Proof.

Sets are added to in lines 1 and 1. First, consider that was added to in line 1. Let . Since is an extremal mp-subgraph, there is an mp-subgraph in such that . Let be the connected component of containing . Then there is a -path in . The concatenation of with a -path of is a tolled -walk containing . Since , it holds that .

Now, we consider that was added to in line 1. Let , , and for be obtained in lines 1,  1, and 1, respectively, of the same iteration that was obtained. Observe that every vertex has a non-neighbour in because otherwise would be a clique, which would mean that is contained in some mp-subgraph of , and then would belong to by the construction of . For any vertex , we will denote by a vertex of that is not adjacent to . Observe that because otherwise it would exist an mp-subgraph outside containing . This implies that .

For every toll concave set of type 1 added to , we associate a natural number . It is clear that a set is added to at most once and this occurs in lines 1 or 1. If is added to in line 1, set . If is added to in line 1 and all members of , of the same iteration, are not toll concave, set . Otherwise, define and is toll concave.

It is clear that is well defined. We use induction on to prove that for every . For the basis, consider . One can choose for any because, for any set , there is no edge between a vertex of and a vertex of . We will show that we can always do at least one of the following choices for .

Now, suppose that Choice 2 is possible. Let be a -path of , let be a -path of , and let . It is clear that the paths , , and form a tolled -walk . For every , it holds that because . Since induces a connected graph by Lemma 3.6 and every vertex of has a neighbor in , Lemma 2.1(2) implies that for every .

Next, we show that . Let be the vertex sets of the connected components of . Since and are connected graphs, for , every contains a neighbor and a neighbor . Now, let be a -path of and be a -path of . Now, for each set , the paths and can be used to find a tolled -walk such that, using Lemma 2.1(2), we can conclude that .

For this case, it remains to show that . Since is not toll concave, there are vertices for which there is a tolled -walk containing some vertex of . At least one of belongs to because is toll concave. On one hand, both belong to and we can write and for . We claim that if , then . Then, suppose that and . This means that is not a clique. Hence, contains but not contains , i.e., has a non-neighbor in . This implies that and , a contradiction, because . Therefore, either and or . In the latter case, since every vertex of contains a neighbor in , we can assume that and, analogously, that . On the other hand, we can write and for . As in the previous case, we can assume that and, since every vertex of has a neighbor in , we can assume that . In both hands, using Lemma 2.1(2), we have that .

We now consider that only Choice 3 can be done. This implies that the only member of containing is and all vertices of the other members are universal to . The proof that is the same of the previous case. Since any is not toll concave, there are vertices for which . At least one of , say , belongs to because is toll concave. Hence, there is a tolled -walk or a tolled -walk containing vertices of . Since induces a connected graph by Lemma 3.6, by Lemma 2.1(1), we have .

Now, consider and that the result holds for every toll concave set added to such that . This means that contains at least one member that is toll concave of type 1, say . By the induction hypothesis, there is such that for every vertex of , in particular for .

We claim that some vertex of has a non-neighbor in . Suppose the contrary. Since is not toll concave, there exist vertices such that