A Note on the Maximum Number of Minimal Connected Dominating Sets in a Graph

11/11/2021
by   Faisal N. Abu-Khzam, et al.
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We prove constructively that the maximum possible number of minimal connected dominating sets in a connected undirected graph of order n is in Ω(1.489^n). This improves the previously known lower bound of Ω(1.4422^n) and reduces the gap between lower and upper bounds for input-sensitive enumeration of minimal connected dominating sets in general graphs as well as some special graph classes.

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1 Introduction

A connected dominating set in a graph is a set of vertices whose closed neighborhood is that induces a connected subgraph. A connected dominating set is inclusion minimal if it does not contain another connected dominating set as a proper subset.

Enumerating all minimal connected dominating sets in a given graph can be trivially performed in . Whether a better enumeration algorithm exists was one of the most important open problems posed in the first workshop on enumeration (Lorentz Center, Netherlands, 2015) [2]. The problem has been subsequently addressed in [5] where an algorithm that runs in was presented. This slightly improves the upper bound on the number of minimal connected dominating sets in a (general) graph.

On the other hand, the maximum number of minimal connected dominating sets in a graph was shown to be in [3]. This lower bound is obviously very low compared to the upper bound and to the running time of the current asymptotically-fastest exact algorithm, which is in [1]. The gap between upper and lower bounds is narrower when it comes to special graph classes. On chordal graphs, for example, the upper bound has been recently improved to [4]. Other improved lower/upper bounds have been obtained for AT-free, strongly chordal, distance-hereditary graphs, and cographs in [3]. Further improved bounds for split graphs and cobipartite graphs have been obtained in [6].

In this note we report an improved lower bound on the maximum number of minimal connected dominating sets in a graph. This is related to the enumeration of all the minimal connected dominating sets since it also gives a lower bound on the asymptotic performance of any input-sensitive enumeration algorithm.

2 Graphs with Large Minimal Connected Dominating Sets

Given arbitrary positive integers , we construct a graph of order as follows.

The main building blocks of consist of copies of a base-graph , of order . The vertex set of consists of three layers. The first layer is a set that induces a clique. The second is an independent set , while the third layer consists of a singleton . Each vertex has exactly neighbors in : . In other words, the base-graph has a maximum anti-matching111An anti-matching in is a collection of disjoint non-adjacent pairs of its vertices. . In fact, induces a copy of minus a perfect matching. Finally the vertex is adjacent to all the vertices in . Figure 1 below shows the graph for .

Figure 1: The graph
Lemma 1.

For each , the graph has exactly minimal connected dominating sets that have non-empty intersection with the set .

Proof.

The set cannot have more than two vertices in common with any minimal connected dominating set, since any two elements of dominate . Any minimal connected dominating set that contains exactly one vertex of must contain the vertex , to dominate , and one of the neighbors of (to be connected). There are sets of this type. Moreover, each pair of elements of dominates . So a minimal connected dominating set can be formed by (any) two elements of and any of the elements of (to dominate ). There are such sets. ∎

The hub-vertex in must be in any connected dominating set, being a cut-vertex. Therefore, there is no need for the set in to induce a clique (in ), being always dominated by . In other words, the counting used in the above proof still holds if each copy of is replaced by in . Here denotes the set of edges connecting pairs of vertices in . The below figure shows without the edges between pairs of element of in each copy of .

Figure 2: The graph
Theorem 1.

The maximum number of minimal connected dominating sets in a simple undirected connected graph of order is in .

Proof.

By Lemma 1, each copy of the graph has minimal connected dominating sets. There are such graphs in , in addition to the vertex that connects them all. Every minimal connected dominating set must contain and at least one element from in each . Therefore, the total number of minimal connected dominating sets in is . The claimed lower bound is achieved when , which gives a total of . ∎

We note that is a -degenerate graph that is also bipartite (since the set in each copy of can be an independent set). Furthermore, we observe that is planar. To see this, simply re-order the elements of in each copy of as shown in Figure 3 below.

Figure 3: A plane drawing of

Therefore, we can obtain an improved lower bound for 3-degenerate, bipartite and planar graphs. We conclude with the following corollary.

Corollary 1.

The maximum number of minimal connected dominating sets in a 3-degenerate bipartite planar graph of order is in .

3 Conclusion

The method we adopted for constructing asymptotic worst-case examples for enumerating minimal connected dominating sets consists of combining copies of a certain base-graph having a particular subset of vertices that must contain elements of any minimal connected dominating set, being linked to a main hub-vertex. For example, the graph has 36 minimal connected dominating sets, each of which must contain elements of the set , which in turn is linked to the hub-vertex in .

The main question at this stage is: can we do better? We believe it is very difficult to find a base-graph of order eight or less that can be used to achieve a higher lower-bound since it would have to have at least 25 minimal connected dominating sets. Moreover, any better example that contains more than 9 vertices must have a much larger number of minimal connected dominating sets. For example, to achieve a better lower bound with a base-graph of order 10 (or 11), such a graph must have at least 54 (respectively 80) minimal connected dominating sets. It would be challenging to obtain such a construction, which is hereby posed as an open problem.

References

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  • [2] H. L. Bodlaender, E. Boros, P. Heggernes, and D. Kratsch, Open problems of the Lorentz workshop “Enumeration Algorithms using Structure”, Utrecht University Technical Report UU-CS-2015-016, 2015.
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