Neighborhood preferences for
minimal dominating sets enumeration
Abstract
We investigate two different approaches to enumerate minimal dominating sets of a graph using structural properties based on neighborhood inclusion. In the first approach, we define a preference relation on a graph as a poset on the set of vertices of . In particular, we consider the poset of closed neighborhood inclusion and define the notion of preferred dominating set as dominating sets that correspond to minimal ideals of . We show that graphs with a unique preferred dominating set are those who are dominated by simplicial vertices and show that there is a polynomial delay algorithm to enumerate minimal dominating sets if there is one to enumerate preferred dominating sets. In the second approach we consider intersections of minimal dominating sets with redundant vertices, i.e., vertices that are not minimal in . We show in a generalized class of split graphs that there is a linear delay algorithm to enumerate minimal dominating sets if these intersections form an independent set system. Graphs that share this property include completed free chordal graphs which improves results from [14] on free chordal graphs.
Keywords:
Graphs, Dominating sets, Enumeration algorithms, Preferred enumeration1 Introduction
We consider the problem of enumerating all (inclusionwise) minimal dominating sets of a given graph, called the DomEnum problem. A dominating set in a graph is a set of vertices such that every vertex of is either in or is adjacent to some vertex of . It is said to be minimal if it does not contain any other dominating set as a subset. We say that an enumeration algorithm is outputpolynomial if its running time is bounded by a polynomial depending of the sum of the sizes of the input and the output. An algorithm is said to be incremental polynomial if the running time between two outputs is bounded by a polynomial depending of the size of the input and already outputted solutions. Finally, we say that an algorithm is polynomial delay if the running time between two outputs is bounded by a polynomial depending of the size of the input. Note that the existence of a polynomial delay algorithm gives an incremental polynomial algorithm which gives an outputpolynomial algorithm. For further details on enumeration problems complexity, see [3, 12].
To date, the DomEnum problem is still open for general graphs. Recently, it has been proved in [14] that this problem is equivalent to the problem of enumerating all minimal transversals of a hypergraph, called the TransEnum problem. The best known algorithm is output quasipolynomial and comes from hypergraph minimal transversals enumeration [9]. However, several classes of graphs are known to admit outputpolynomial algorithms. For example, it has been shown that there exist incremental polynomial algorithms for chordal bipartite graphs or graphs of bounded conformality [7, 11]. Polynomial delay algorithms are known for degenerate, line, chordal and strongly chordal graphs [5, 6, 15, 16]. Linear delay algorithms are known for split, free chordal, permutation, interval graphs and graphs with bounded clique width [2, 13, 14].
We investigate two different approaches to enumerate minimal dominating sets of a graph using structural properties based on neighborhood inclusion. In the first approach, we define a preference relation on a graph as a poset on the set of vertices of . In particular, we consider the poset of closed neighborhood inclusion and define the notion of preferred dominating sets of as minimal dominating sets that correspond to minimal ideals of . We show that graphs that have a unique preferred dominating set are those who are dominated by their simplicial vertices. We then rewrite a result from [8] on implicational systems to show that there exists a polynomial delay algorithm to enumerate minimal dominating sets if there is one to enumerate preferred dominating sets. In the second approach we consider intersections of minimal dominating sets with redundant vertices, i.e., vertices that are not minimal in . We show in a generalized class of split graphs that there is a linear delay algorithm to enumerate minimal dominating sets if these intersections form an independent set system. Graphs that share this property include completed free chordal graphs which improves results from [14] on free chordal graphs.
The rest of the paper is organized as follows. In the next section, we give preliminary notions and introduce new notations. In sections 3 and 4, we focus on the first approach and show the existence of a polynomial delay algorithm to enumerate minimal dominating sets from preferred dominating sets. In sections 5 and 6, we focus on the second approach and give a linear delay algorithm to enumerate minimal dominating sets on a generalized class of split graphs. We conclude the paper by discussing the outlooks of this work.
2 Preliminaries
We refer to [4] for graph terminology not defined below; all graphs considered in this paper are undirected, finite and simple. A graph is a pair , where is the set of vertices and is the set of edges. A clique is a graph in which every two vertices are adjacent. When considering functions that depend on the size of a graph, we usually use and for the size of and . An edge between and is denoted by (or ) instead of . The subgraph of induced by , denoted by , is the graph ; is the graph . We note (or simply if there is no ambiguity) the set of neighbors of in defined by ; is the set of closed neighbors defined by . We say that a vertex is simplicial if its neighborhood form an induced clique. We note the number of neighbors of , i.e., . For a given , we respectively write and the sets defined by and .
A dominating set in a graph is a set of vertices such that every vertex of is either in or is adjacent to some vertex of . It is said to be minimal if it does not contain any other dominating set as a subset. The set of all minimal dominating sets of is denoted by . Let be a dominating set of and . We say that has a private neighbor in if . Note that a private neighbor of a vertex in is either itself or a vertex in . The set of private neighbors of in is denoted by ; is a minimal dominating set if and only if for every . In this paper, we restrict ourselves to connected graphs which has no impact on dominating sets enumeration since disconnected subgraphs can be considered separately.
A partial order (or poset) on a set is a binary relation on which is reflexive, antisymmetric and transitive, denoted by . Here, is called the ground set of . Let be two elements of , if or , then and are said comparable, otherwise they are said incomparable. A subset of a poset in which every pair of elements is comparable is called a chain. A subset of a poset in which no two distinct elements are comparable is called an antichain. A set is said to be an ideal if and implies . For an element we associate the principal ideal . The filter of is the dual . The set of all ideals of is denoted by . For a subset , we denote by (resp. ) the maximal (resp. minimal) elements of with respect to .
2.1 Preference relation
We define a preference relation on as a partially ordered set on the set of vertices of , which compares couples of vertices , where reads is preferred to . The most known preference relations are the lexicographical order, where any two vertices are comparable, and indifference order or antichain, where no two vertices are comparable. In this paper, we define as the poset of closed neighborhood inclusion of , with ground set and where if . We say that two vertices and are twins if they share the same closed neighborhood, i.e., . These twins are similar in and thus ignored by keeping only one. In fact, it is well known that twins play no role in the complexity of minimal dominating sets enumeration: one can identify and remove them in polynomial time preprocessing, and then generate minimal dominating sets and permute twins at each output to obtain all solutions. In the following, all graphs are considered without twins.
It is well known that the powerset ordered by inclusion is a boolean lattice in which forms an antichain (i.e., a clutter or simple hypergraph). The motivation behind the use of a preference relation is to reduce the space of solutions from this boolean lattice to a distributive lattice. In fact, it can be seen that each comparability removes from the boolean lattice given by the interval . The use of as preference relation guarantees that minimal dominating sets are antichains of and thus that they do not belongs to such intervals. We give simple but essential properties on such preference, together with an example of a graph and its preference relation in Figure 1.
Lemma 1
If , then is an antichain of .
Proof
Suppose is not an antichain of . Then there exist such that , and thus . Then is empty, which is a contradiction with being a minimal dominating set. ∎
Lemma 2
is always a dominating set.
Proof
Let . Then there exists such that and thus . In particular, and thus is connected to . ∎
Fact 2.1
For all , there exists a minimal dominating set containing .
Proof
Note that is a dominating set. Thus there is a minimal dominating set containing . ∎
Lemma 3
If is simplicial, then and for all .
Proof
If is simplicial, then forms a clique and therefore for all . Since has no twins, then for all . ∎
2.2 Graph completion
We recall the graph completion introduced by Kanté et al. in [14]; this completion adds edges to the graph by keeping its set of minimal dominating sets invariant. Using same notations as in [14], we note the set of socalled redundant vertices defined by and the set of socalled irredundant vertices defined by .
Definition 1 ([14])
The completion graph of a graph is the graph with vertex set and edge set , i.e., is obtained from by adding precisely those edges to that make into a clique.
Proposition 1 ([14])
For any graph G, .
Proposition 2
For any graph G, .
Proof
By Def. 1, for every . Thus, if then there exists such that and therefore . Now, if then for every and thus . ∎
2.3 Preferred dominating sets
It is well known that for a poset , the set of ideals of is in bijection with the set of antichains of . From Lemma 1, we deduce that for and for any minimal dominating set there is a unique corresponding ideal with . But the converse is not true, since the empty set is always an ideal. We say that an ideal is a dominating ideal if is a dominating set of , and that is a minimal dominating ideal if is a minimal dominating set of . A dominating set is called preferred whenever and dominates implies that . It is clear that a preferred dominating set is a minimal dominating set. Moreover, for every minimal dominating set , there exists a preferred dominating set such that . Now, note that if any subset dominates , then dominates . In particular, every ideal that dominates is a dominating ideal. This is an important property of our neighborhood preference.
Given a graph , we denote by the set of minimal dominating ideals of , ordered by inclusion, and note the set of preferred ideals of . We denote by the set of preferred dominating sets defined by . Figure 3 shows the bijection between minimal dominating sets and ideals given by Lemma 1, using the same graph as in Figure 1.
3 Graphs with a unique preferred dominating set
In this section, we are interested in graphs with a unique preferred dominating set. We show that if a graph has a unique preferred dominating set, then it is directly obtained by minimal elements of . We then give a simplicial characterization of such graphs and show that they include free chordal graphs.
Lemma 4
If a graph has a unique preferred minimal dominating set , then . Moreover, is a maximal independent set of .
Proof
Suppose and . By Lemma 2, we have since is a dominating set. Then there exists and thus is a dominating set. Moreover, contains at least one preferred dominating set that contains since . Thus this contradicts that . Suppose that is not an independent set. Let such that and let . Because , . Thus since is a private neighbor of . Then and which contradicts the fact that . We conclude that is a maximal independent set since is a minimal dominating set. ∎
Note that the converse of Lemma 4 is not true for the graph given in Figure 2: is a preferred dominating set but also is. However, an important consequence of this lemma is that if has a unique preferred dominating set, then it is directly given by . In the following, we give a characterization of graphs and completion graphs that have a unique preferred dominating set.
Lemma 5
A vertex is simplicial if and only if is not a dominating set.
Proof
Let us consider any simplicial vertex and show that is not a dominating set. By Lemma 3, for all . Hence, and does not dominate . Conversely, let be a vertex such that is not a dominating set. By Lemma 2, . Suppose that is not simplicial, then there exist such that . Thus and and so . Moreover, are connected to every since for all such and . We conclude that is a dominating set which is absurd. ∎
Theorem 3.1
A graph has a unique preferred dominating set if and only if every vertex in is simplicial.
Proof
Suppose that has a unique preferred dominating set, i.e., . By Lemma 4, . Suppose that there exists such that is not simplicial. By Lemma 5, is a dominating set and there exists a preferred dominating set such that . Since we conclude that which is absurd. Conversely, suppose that all minimal vertices are simplicial. Then by Lemma 2, is a dominating set. Moreover, is minimal and is the only preferred dominating set since by Lemma 5, is not a dominating set for all . ∎
Corollary 1
A graph has a unique preferred dominating set if and only if the set of simplicial vertices is a dominating set.
The following proposition shows that such graphs include free chordal graphs, which strictly include split graphs. However, note that our characterization only relies on simplicial vertices; big induced cycles are not forbidden as long as they are dominated by simplicial vertices.
Proposition 3
If is free chordal then it has a unique preferred dominating set.
Proof
Let be a free chordal graph and suppose that there exists at least one minimal vertex that is not simplicial. Then has at least two disconnected neighbors and . As is minimal, both and have at least one neighbor not in . We note and such neighbors of and . As the graph is chordal, note that or otherwise it would create an induced . Therefore, vertices forms an induced , which is against the hypothesis. ∎
We end this section with a characterization of graphs for which the completion graph has a unique preferred dominating set. Such graphs are those who are completed to a split graph, which were shown in [14] to include free chordal graphs.
Theorem 3.2
Let be any graph, the three following statements are equivalent:

is dominated by vertices that are simplicial in ,

has a unique preferred dominating set,

is a split graph.
Proof
If is dominated by vertices that are simplicial in then by Corollary of Corollary 1, has a unique preferred dominating set. If has a unique preferred dominating set then by Theorem 3.1 every vertex is simplicial and thus is a split graph. Now, if is a split graph, then it is dominated by its set of simplicial vertices and thus is dominated by vertices that are simplicial in . This conclude the proof. ∎
4 Enumeration from preferred dominating sets
In this section, we show using results from Ennaoui’s thesis [8] on generating keys of a database that there exists a polynomial delay algorithm to enumerate if there is one to enumerate . We reformulate and reproved these results in our context for greater clarity. This motivates the study of preferred enumeration.
Lemma 6 ([8])
Let . There exists and such that .
Proof
Since is not preferred, there exists such that is still a dominating ideal. Since is a minimal dominating set, every has a private such that . Then, there exists a minimal dominating set such that and . In other words, is equal to plus a subset of predecessors of . Therefore and . ∎∎
A consequence of Lemma 6 is that every minimal dominating set that is not preferred is accessible from a minimal dominating set by adding one vertex to and by removing its predecessors in . It thus gives the existence of a transition graph where nodes are elements of and where there is an edge from to if for some . The idea behind this transition graph notion, as stated in [10] by Gely et al., is to define a rooted spanning tree on this transition graph such that any algorithm that search the graph using this tree will enumerate all . This method was also known as reverse search in [1]. Note that in the case of several preferred dominating sets we are interested in a spanning forest. Lemma 7 defines such spanning forest with for roots nodes and Theorem 4.1 shows how it can be used to get a polynomial delay algorithm for minimal dominating ideals enumeration.
Lemma 7 ([8])
For all , there exists a unique , denoted by , satisfying the following:

where is the lexicographically largest element in such that is a dominating set.

is the lexicographically largest minimal dominating set contained .
Proof
Let , is a dominating set and for some . Since , is not empty and by Lemma 6 there exist and such that and thus is not empty. Then, for a given , we consider the lexicographically largest such that , and the lexicographically largest ideal in . This defines the relation. ∎
Theorem 4.1 ([8])
There is polynomial delay and space algorithm to enumerate if there is one to enumerate .
Proof
Suppose that there is a polynomial delay and space algorithm to generate and let be the list of preferred ideals generated by the algorithm. Let us consider the th step, when is outputted. Using Lemma 6, we search the rooted tree at using a depthfirst algorithm in polynomial delay and space, before continuing to . The parent relation can be computed in polynomial time and ensures that rooted trees are disjoint and solutions only generated once. ∎
As a consequence, knowing how to generate in polynomial delay and space is sufficient to generate in polynomial delay and space. Thus, this gives a polynomial delay algorithm to enumerate whenever or its completion graph has a unique preferred dominating set, since is directly obtained by on these graphs. However, we show using a second approach for generating minimal dominating sets using neighborhoods that there exists a linear delay algorithm for a generalized class of split graphs. Such graphs include split graphs and thus (completion) graphs that have a unique preferred dominating set.
5 Bipartitions of minimal dominating sets
In this section, we investigate an approach for enumerating minimal dominating sets from their intersection with redundant vertices. For any dominating set , we note its subset of redundant vertices and its subset of irredundant vertices. Then and form a bipartition of . We note the set defined by .
We call a graph quasisplit if the vertex set can be partitioned into one clique and a set of disconnected cliques such that every vertex of has a neighbor in and none in , for all . Note that for a given graph, is uniquely determined. Clearly, is a split graph if for all . Note that quasisplit graphs contain cobipartite incidence graphs that were used to polynomially reduce minimal dominating sets enumeration to minimal transversals in hypergraphs in [14]. Thus, enumerating minimal dominating sets of a quasisplit graph is harder than minimal transversals. We now recall some results from Mary’s thesis [17] on a generalization of graph twins for minimal dominating sets.
Definition 2 ([17])
Two connected vertices and are called dequivalent if implies that . A class of dequivalent vertices is a set such that are dequivalent for all ; then is a clique.
Proposition 4 ([17])
Two vertices and are dequivalent if and only if , where is the set of minimal dominating sets containing .
Note that if is quasisplit then every clique , , is a set of dequivalent vertices since for all . We show for such graphs that is equal to the set of every subset for which all have a private neighbor in some .
Lemma 8
Let be a quasisplit graph and . If every has a private neighbor in some , , then for every .
Proof
Let and . Note that is a dominating set. Indeed, if , then by Lemma 2, is a dominating set of . Since , , are sets of dequivalent vertices, is a dominating set of and so is . If , then and by construction of , for all such that . Thus, is a dominating set of . We now show that every has a private neighbor. If , then for some , , and ; thus has a private neighbor. Now, since every has a private neighbor in some , , then is not connected to any vertex of and thus . ∎
Lemma 9
Let be a quasisplit graph and . Then every has a private neighbor in some , , and .
Proof
Let , we first show that every vertex in has a private neighbor in some , . Suppose that there exists such that every verifies for some . Since dominates and , and thus which is absurd. We now show that . Since , intersects at least every clique , such that . Moreover, does not intersect any clique , such that since no vertex in can have private neighbors as and thus . At last, for every since every is a set of dequivalent vertices. We conclude that . ∎
Theorem 5.1
If is quasisplit then has a private neighbor in some .
Proof
Corollary 2
Let be a quasisplit graph. Then is an independent set system if and only if implies that for all , has a private neighbor in some .
In the following, we call ISquasisplit a quasisplit graph on which the set is an independent set system and show that such graphs include completed free chordal graphs.
Lemma 10
If is a connected free chordal graph then every two connected irredundant vertices are dequivalent.
Proof
Let be two connected vertices that are not dequivalent. By definition there exists such that . Without loss of generality, let us choose (rather than ) to have such neighbor . Note that and since and thus there exist such that and . Now, since , there exist such that and . Since is chordal, forms an an induced which is excluded. ∎
Theorem 5.2
If is a connected free chordal, then is ISquasisplit.
Proof
Let and be the connected component of . By Lemma 10, are disconnected cliques for every . Thus, is quasisplit. Let us suppose that is not ISquasisplit. Using Corollary 2, there exist and such that every belongs to a clique . Let be one such vertex and let us note the clique , , in which belongs. Then since . Moreover, since there exists such that . In particular . Note that as then and thus there exists such that . Now, since there exists such that . We note the clique , , in which belongs. Since and , . Then by hypothesis, and therefore is connected to some vertex which is connected to as . Note that since , is not connected to . At last, since then and thus there exists some . Since is chordal and since every two vertices at distance two in path are disconnected, this leads to an induced in which is excluded as is free. ∎
One question remains to know if there is a polynomial time algorithm to recognize quasisplit graphs.
6 Enumeration algorithm
We now describe an algorithm which takes a quasisplit graph , a linear ordering of , and enumerate in linear delay whenever is an independent set system. At each step of the algorithm, given , the algorithm computes the largest ^{1}^{1}1 if . and check if can be extended by adding some candidate such that . The recursive tree stops when cannot be extended, that is if no minimal dominating set intersects on .
Theorem 6.1
Algorithm 1 enumerates in linear delay and polynomial space on quasisplit graphs whenever is an independent set system.
Proof
We prove the completeness of the algorithm using induction on the number of elements in . First, by Lemma 2, belongs to since is a dominating set and . At first call, is outputted (Line 1). Assume now that every such that has been outputted by the algorithm and let such that . Let and be the vertex such that . Since is an independent set system, there exists such that . Now, by inductive hypothesis since , is outputted by the algorithm. Also is greater that every vertex in with respect to (Line 3). By Theorem 5.1, since , has a private neighbor in some such that and thus (Line 4). Therefore is outputted by the call (Line 5) of Enum().
Now, if a set is outputted by the algorithm, then either (Line 1) or every vertex has a private neighbor in some clique such that (Line 4). By Theorem 5.1, .
We now analyze the complexity of the algorithm. Note that computing (Line 2) and such that (Line 3) takes operations. We show using arrays that testing in the loop if there exists such that and (Line 4) is bounded by . Before calling the algorithm, compute an array of size such that for every . This array will be used to know the number of remaining vertices to dominate in each clique . Also, compute an array of size such that if . Using these two arrays, one can access in constant time to the number of remaining vertices to dominate in the unique clique in which belongs by calling . Finally, consider an array of size initialized to zero. This array will be used to know if a vertex is dominated by , by setting if is connected to some and if . Note that this preprocessing takes at most steps. Now, at each iteration of the loop (Line 3), when considering a new candidate , do the following : for each , update and whenever . Note that is decreased to zero if and only if verifies and for . This testing takes at most steps. When backtracking , undo the changes by setting and for every such that . This also takes at most . Since is bounded by , the whole loop (Line 3) takes at most steps. ∎
As a consequence, there exists a linear delay algorithm to enumerate minimal dominating sets in free chordal graphs after polynomial preprocessing to compute , and thus the partition . Indeed, at each output of Algorithm 1, one just has to extend to for every set in the cartesian product of all that can be computed in constant delay.
Also, note that Algorithm 1 can be extended to other set system structures of .
An accessible set system is a family of set in which every nonempty set contains an element such that belongs to the family. We obtain the same result as in Theorem 5.1 for accessible set systems, i.e., is an accessible set system if implies that there exists such that for all , has a private neighbor in some . For such graphs, we can enumerate in polynomial delay by testing every candidate at each step to extend using same techniques as in Section 4.
For future research, we are interested in structural properties of and in an extension of the clique restriction on quasisplit to other simple class of graphs such as cographs, split graphs, etc.
7 Conclusion
In this paper, we investigated two different approaches to enumerate minimal dominating sets of a graph using structural properties based on neighborhood inclusion. We introduced preferred enumeration and showed that there is a polynomial delay algorithm to enumerate minimal dominating sets if there is one to enumerate preferred dominating sets. We then studied intersections of minimal dominating sets with redundant vertices and gave a linear delay algorithm to enumerate minimal dominating sets in a generalized class of split graphs whenever such intersections form an independent set system. We showed that such graphs include completed free chordal graphs which improves results from [14] on free chordal graphs.
We highlight two directions for future fundamental research in enumeration problems: (1) preferred enumeration may be of interest in some applications like data mining where the size of the output is usually exponential, (2) preferred enumeration may reveal structural properties of graphs and discrete structures to obtain more efficient algorithms. Indeed, more structural properties can be captured by the preference relation as shown in this paper for graphs with unique preferred dominating set.
Acknowledgment:
This work has been funded by the ANR project Graphen no. 398 / PNE / ENS / FRANCE / 20152016 and the CNRS Mastodons project Qualisky 20152017.
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