## 1 Introduction

A *convexity* of a graph is a pair where is a family of subsets of satisfying all the following conditions: , , and is closed under intersections (i.e., for each ). Each set of the family is called *-convex*. Distinct convexities of a graph have been widely studied in the last years. Several articles can be found in the literature dealing with algorithmic and complexity issues of parameters related to different kind of convexities. Interesting enough is to compare the behavior of these parameters under different convexities from a computational complexity perspective. For an introduction to the different parameters studied in the literature related to a convexity of a graph, see e.g. Douchet1988 .

A family of -convex sets, each of which different from , is a *-convex -cover* if and a *-convex -partition* if, in addition, for each with . In this work, we consider two associated decision problems: in one case, the input is just a graph and, in the other case, the input is a graph and an integer . In both cases, the decision problem is: ‘Does the graph have a -convex -cover (resp. -convex -partition)?’.

All graphs in this work are finite, undirected, without no loops and no multiple edges. Let be a graph. We denote by and the vertex set and the edge set of , respectively. The neighborhood of a vertex of is denoted by , and stands for the set . If is a set, denotes its cardinality. The *degree* of a vertex of is . If is a subset of , then the set of those vertices of with at least one neighbor in is denoted by , and stands for . The *length* of a path is its number of edges. The *distance* between two vertices and of , denoted by , is the minimum length of a path having and as end-vertices. The *diameter* of a graph is the maximum distance between two of its vertices. A path is *induced* if there is no edge joining two nonconsecutive vertices of the path. We denote by the induced path on three vertices. We denote the *complement graph* of by . Graph is *co-bipartite* if is bipartite. For any other graph-theoretic notions not given here, see West01 .

Let be a set of paths in and let . If and are two vertices of , then the *-interval* is the set of all vertices lying in some path having and as its end-vertices. Let . Let be the family of all sets of vertices of such that for each path whose end-vertices belong to , every vertex of also belongs to ; i.e., consists in those subsets of such that . It is easy to show that is a convexity of and is called the *path convexity generated by *. Some of the most studied path convexities are the *geodesic convexity*, the *monophonic convexity*, the *-convexity*, and the *-convexity*, which are the convexities whose convex sets are generated by the set of all minimum paths, the set of all induced paths, the set of all paths of length three, and the set of all induced paths of length three of the graph, respectively.

A set of vertices of a graph is *digitally convex* if, for each vertex of , implies . The *digital convexity* of a graph is the pair where is the set of all digitally convex sets of . This convexity was introduced in PR-digitalconvexity-1966 as a tool to filter digital images.

The problem of deciding whether has a -convex -partition was introduced by Artigas et al. paper2 . They proved that the problem is NP-complete for each fixed integer , under the geodesic convexity. Centeno et al. paper1 proved that the problem is also NP-complete under the -convexity if a graph and an integer are given as input. The problem of deciding whether a graph has a -convex -cover was introduced in paper3 also by Artigas et al. This problem is NP-complete under the geodesic convexity for each fixed integer paper3 ; paper4 .

This article is organized as follows. In Section 2, we deal with the problems of covering with convex sets and of partitioning into convex sets a graph under the digital convexity. In Section 3, we present results in connection with the problem of partitioning a graph into convex sets under the -convexity, as well as the -convexity, and we also consider the problem of covering a graph with -convex sets. In Section 4, we address these problems under the monophonic convexity.

## 2 Digital convexity

We will call *d-convex* to any convex set under the digital convexity. Our first result characterizes d-convex sets as the complements of the closed neighborhoods of sets of vertices.

###### Proposition 1

Let be a graph, , and . For each vertex of , if and only if .

In fact, is equivalent to which, in turn, is equivalent to .

###### Lemma 2

A set of vertices of a graph is d-convex if and only if for some set .

Suppose that is a d-convex set of and let . We claim that . On the one hand, if , then and, by virtue of Proposition 1, . On the other hand, if , then, by Proposition 1, and, since is d-convex, . We conclude that , as desired.

Conversely, let and . Let such that . Notice that (in fact, if there were some vertex , then there would be some vertex , a contradiction). Thus, or, equivalently, . Hence, . This proves that is a d-convex set of .

A *total dominating set* of a graph is a set of vertices of such each vertex of has at least one neighbor in . Our next result shows that the existence of a covering with few digitally convex sets is equivalent to the existence of sufficiently small total dominating sets in the complement.

###### Theorem 3

A graph has a d-convex -cover if and only if has a total dominating set of cardinality at most .

Let be a total dominating set of with . By virtue of Lemma 2, is a d-convex set for each . We claim that is a cover of . Since is a total dominating set of , for each in , there exists some such that . Therefore, .

Conversely, let be a d-convex -cover of . Hence, by Lemma 2, for some . Choose a vertex for each . Notice that it may happen that even if . Let . By construction, . As is a cover of , for each there is some such that . This proves that is a total dominating set of with at most vertices.

Since the problem of deciding, given a graph and a positive integer , whether has a total dominating set of size at most is NP-complete td , Theorem 3 implies the following.

###### Corollary 4

It is NP-complete to decide, given a graph and an integer , whether has a d-convex -cover.

Notice that Corollary 4 proves the NP-completeness of deciding whether a graph has d-convex -cover when is part of the input. However, the complexity for the case in which is a fixed integer is unknown.

We were not able to determine the computational complexity of the problem of deciding, given a graph and an integer , whether the graph has a d-convex -partition. Below, we present a result that shows that if the graph is assumed bipartite and we fix , then the problem becomes polynomial-time solvable. We first prove the following lemma.

###### Lemma 5

Let be a connected graph. If has a d-convex -partition, then has diameter at least 3.

Let be a d-convex -partition of a connected graph . Since is connected, there exists for each such that is adjacent to . Since (where sums should be considered modulo 2) and is a -convex set, there exists some vertex adjacent to such that . By construction, is an induced path of length three of . Consequently, the diameter of is at least .

We are now ready to prove the following result.

###### Theorem 6

A bipartite graph has a d-convex -partition if and only if has diameter at least 3. Moreover, if such a partition exists, it can be found in polynomial time.

If is disconnected and is any connected component of , then is clearly a d-convex -partition of that, in addition, can be built in polynomial time. Hence we assume, without loss of generality, that is connected. We have already proved in Lemma 5 that if is a connected graph having a d-convex -partition, then has diameter at least . Conversely, assume now that is a connected bipartite graph with bipartition and diameter at least 3. Two vertices such that and the two sets and can be computed polynomial time. It is not hard to see that is a d-convex -partition of .

## 3 Convexities generated by paths of length three

We1 call *-convex* (resp. *-convex*) to any convex set under the -convexity (resp. -convexity). Let be a bipartite graph with a bipartition and let be an integer such that . We construct a bipartite graph as follows: we take a copy of and a complete bipartite graph where , we add an edge connecting each vertex in with a vertex of one of the partite sets of , and we add an edge connecting each vertex in with a vertex of the other partite set of , so that there are no two vertices in the copy of in adjacent to the same vertex in the complete bipartite graph .

A *cut* of a graph is a partition of into two sets and , denoted by . The set of all edges having one endpoint in and the other one in is called the *edge cut* of the cut . A *matching cut set* is an edge cut that is a (possible empty) matching (i.e., no two edges of the edge cut share an endpoint).

It can be easily proved that the problem of deciding whether a graph has a -convex -partition and the problem of deciding whether a graph has matching cut set are equivalent.

###### Lemma 7

A graph has a -convex -partition if and only if has a matching cut set.

###### Lemma 8

Let be an integer such that . A graph has a -convex -partition if and only if has a -convex -partition.

If has a -convex -partition , then has a -convex -partition , where is the set of vertices of the copy of in .

Conversely, suppose that has a -convex -partition . Let be the vertex set of the copy of in and let be the vertex set of the copy of in . We claim that for some . Indeed, since , there exist at least two nonadjacent vertices for some and thus (because is -convex). We assume, without losing generality, that ; i.e., . Let . Arguing towards a contradiction, suppose that . Notice that if no vertex in has a neighbor in for some , then is a -convex -partition of the copy in ’ and, in particular, a -convex -partition of , as desired. Hence, we assume, without loss of generality, that there is some vertex adjacent to some vertex for some . By the construction of , there exists a vertex which is adjacent to and nonadjacent to . Consequently, is a path on three vertices of such that but , contradicting the fact that is a -convex set of . This contradiction arose from assuming that . Hence, and thus . Therefore, is a -convex -partition of the copy of in and, in particular, a -convex -partition of .

Since the matching cut set problem is NP-complete even when the input is a bipartite graph LR-2003 , by virtue of Lemma 8, we obtain the following theorem.

###### Theorem 9

For each fixed integer , it is NP-complete to decide, given a graph , whether has a -convex -partition, even if is a bipartite graph.

Notice that, in a bipartite graph, the family of -convex sets in coincides with the family of -convex sets in . Hence, Corollary 9 still holds if ‘-convex’ is replaced by ‘-convex’.

A *stable set* is a set of pairwise nonadjacent vertices. A *clique* is a set of pairwise adjacent vertices. A *split graph* is a graph whose vertex set can be partitioned into an independents set and a clique ; the pair is called a *split partition*.

###### Lemma 10

Let be a split graph with split partition and be an integer such that . If every vertex has degree at least two, then has a -convex -partition if and only if has a -convex -cover.

The ‘only if’ part is clear. Assume that has a -convex -cover . Notice that, for each , because otherwise every vertex would belong to and, since every vertex of is adjacent to at least two vertices of , for every vertex , contradicting . Besides, if and are adjacent, then there is no such that ; otherwise, any other neighbor of in would belong to and so . We have proved that is an independent set for each . Let and let for each . By construction, is a partition. We claim that, for each , is a -convex set. Arguing towards a contradiction, suppose that there exists some and some vertex such that . In particular, and, since is -convex, , contradicting the fact that is an independent set. This contradiction proves the claim. Hence, is a -convex -partition of .

In paper1 , it is proved that it is NP-complete to decide whether a split graph with split partition has a -convex -partition when is part of the input. Looking carefully at the proof given in paper1 , one can readily verify that it is still valid if every vertex in has degree at least two.

###### Theorem 11 ((paper1, , Theorem 2.1))

The problem of deciding, given a split graph with a split partition , where has a -convex -partition is NP-complete, even if each vertex in has at least two neighbors.

Therefore, in virtue of Lemma 10, the result below follows.

###### Corollary 12

It is NP-complete to decide, given a graph and an integer , whether has a -convex -cover, even if is a split graph.

## 4 Monophonic convexity

We will call m-convex to any convex set under the monophonic convexity. We will prove that it is NP-complete to decide, given a graph and an integer , whether has an m-convex -partition, by adapting the proof of (paper3, , Theorem 1) for the analogous result for the geodesic convexity. If is a positive integer, Clique -Partition is the problem of deciding, given a graph , whether the vertex set of can be partitioned into cliques of . It is well known that this problem is NP-complete for all .

###### Theorem 13

It is NP-complete to decide, given a graph and an integer , whether has an m-convex -cover.

Deciding if a set is m-convex can be done in polynomial time paper5 and thus the problem of deciding if a graph has an m-convex -cover belongs to . Let be an instance of the Clique -Partition problem. We assume, without loss of generality, that and is not a complete graph. We construct a graph whose vertex set is and . The proof follows in the same way as the proof of the NP-completeness of Theorem 1 in (paper3, , Theorem 1). It is not hard to prove that any proper m-convex set of is a clique. From this assertion, it follows that has an m-convex -partition if and only if has an -clique partition for some integer such that .

Next, we will prove that deciding whether a graph has an m-convex -cover becomes polynomial-time solvable for . A *clique separator* of a connected graph is a clique such that is disconnected.

###### Lemma 14 (paper5 )

Let be a connected graph, a clique separator of , and the union of the vertex sets of some of the connected components of . Then is an m-convex set of .

###### Corollary 15

If is a connected graph having a clique separator, then has an m-convex 2-cover.

If is a clique separator of and is any connected component of , then, by virtue of Lemma 14, is an m-convex 2-cover of .

Let be a graph. The *m-convex hull* of a set is the inclusion-wise minimal m-convex set of containing or, equivalently, where is the set of all induced paths of . A set of vertices of is an *m-hull set* of if the m-convex hull of is .

###### Lemma 16 (paper5 )

If is a connected graph having no clique separator that is not a complete graph, then every pair of nonadjacent vertices is an m-hull set of .

The following result is an immediate consequence of Lemma 16.

###### Corollary 17

If is a connected graph having no clique separator, then every proper m-convex of is a clique. Moreover, has an m-convex -cover if and only if is a co-bipartite graph.

###### Theorem 18

It is polynomial-time solvable to decide, given a graph , whether has an m-convex -cover.

In order to decide whether a connected graph has an m-convex -cover, we first look for a clique separator in time clique . If a clique separator is found, then, by Corollary 15, has an m-convex 2-cover. If has no clique separator, we test if is co-bipartite, which can be performed in linear time. Consequently, it can be decided in time whether or not a given graph has an m-convex 2-cover.

## References

## References

- (1) P. Duchet, Convex sets in graphs. II. Minimal path convexity, J. Combin. Theory Ser. B 44 (1988) 307–316 (1988).
- (2) D. B. West, Introduction to Graph Theory, Prentice Hall, 2001 (2001).
- (3) A. Rosenfeld, J. L. Pfaltz, Sequential operations in digital picture processing, J. ACM 13 (4) (1966) 471–494 (1966).
- (4) D. Artigas, S. Dantas, M. C. Dourado, J. L. Szwarcfiter, Partitioning a graph into convex sets, Discrete Math. 311 (2011) 1968–1977 (2011).
- (5) C. C. Centeno, S. Dantas, M. C. Dourado, D. Rautenbach, J. L. Szwarcfiter, Convex partitions of graphs induced by paths of order three, Discrete Math. Theor. Comput. Sci. 12 (2010) 175–184 (2010).
- (6) D. Artigas, S. Dantas, M. C. Dourado, J. L. Szwarcfiter, Convex covers of graphs, Mat. Contemp. 39 (2010) 31–38 (2010).
- (7) R. Buzatu, S. Cataranciuc, Convex graph covers, Comput. Sci. J. Moldova 23 (2015) 251–269 (2015).
- (8) R. Laskar, J. Pfaff, S. M. Hedetniemi, S. T. Hedetniemi, On the algorithmic complexity of total domination, SIAM J. Algebraic Discrete Methods 5 (1984) 420–425 (1984).
- (9) V. B. Le, B. Randerath, On stable cutsets in line graphs, Theoret. Comput. Sci. 301 (2003) 463–475 (2003).
- (10) M. C. Dourado, F. Protti, J. L. Szwarcfiter, Complexity results related to monophonic convexity, Discrete Appl. Math. 158 (2010) 1268–1274 (2010).
- (11) S. H. Whitesides, An algorithm for finding clique cut-sets, Inform. Process. Lett. 12 (1981) 31–32 (1981).

Comments

There are no comments yet.