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

1 Introduction

A connected dominating set in a graph $G = (V, E)$ is a set of vertices whose closed neighborhood is $V$ 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 $O (2^{n})$ . 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 $O ((2 - 10^{- 50})^{n})$ 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 $Ω (3^{\frac{n}{3}})$ [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 $O ({1.862}^{n})$ [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 $O ({1.4736}^{n})$ [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 $k, t$ , we construct a graph $G_{t}^{k}$ of order $n = k (2 t + 1) + 1$ as follows.

The main building blocks of $G_{t}^{k}$ consist of $k$ copies of a base-graph $G_{t}$ , of order $2 t - 1$ . The vertex set of $G_{t}$ consists of three layers. The first layer is a set $X = {x_{1} \dots x_{t}}$ that induces a clique. The second is an independent set $Y = {y_{1}, \dots y_{t}}$ , while the third layer consists of a singleton ${z}$ . Each vertex $x_{i} \in X$ has exactly $t - 1$ neighbors in $Y$ : $N (x_{i}) = {y_{j} \in Y : i \neq j}$ . In other words, the base-graph $G_{t}$ has a maximum anti-matching¹¹1An anti-matching in $G$ is a collection of disjoint non-adjacent pairs of its vertices. ${{x_{j}, y_{j}} : 1 \leq j \leq t}$ . In fact, $X \cup Y$ induces a copy of $K_{t, t}$ minus a perfect matching. Finally the vertex $z$ is adjacent to all the $t$ vertices in $Y$ . Figure 1 below shows the graph $G_{t}$ for $t = 4$ .

Figure 1: The graph

G_{4}

Lemma 1.

For each $t > 0$ , the graph $G_{t}$ has exactly $\frac{t^{3} + t^{2}}{2} - t$ minimal connected dominating sets that have non-empty intersection with the set $X$ .

Proof.

The set $X$ cannot have more than two vertices in common with any minimal connected dominating set, since any two elements of $X$ dominate $X \cup Y$ . Any minimal connected dominating set that contains exactly one vertex $x_{i}$ of $X$ must contain the vertex $z$ , to dominate $y_{i}$ , and one of the $t - 1$ neighbors of $x_{i}$ (to be connected). There are $t (t - 1)$ sets of this type. Moreover, each pair of elements of $X$ dominates $Y$ . So a minimal connected dominating set can be formed by (any) two elements of $X$ and any of the elements of $Y$ (to dominate $z$ ). There are $t \frac{t (t - 1)}{2}$ such sets. ∎

The hub-vertex $s$ in $G_{t}^{k}$ must be in any connected dominating set, being a cut-vertex. Therefore, there is no need for the set $X$ in $G_{t}$ to induce a clique (in $G_{t}^{k}$ ), being always dominated by $s$ . In other words, the counting used in the above proof still holds if each copy of $G_{t}$ is replaced by $G_{t} - E (X)$ in $G_{t}^{k}$ . Here $E (X)$ denotes the set of edges connecting pairs of vertices in $X$ . The below figure shows $G_{3}^{3}$ without the edges between pairs of element of $X$ in each copy of $G_{3}$ .

Figure 2: The graph

G_{3}^{3}

Theorem 1.

The maximum number of minimal connected dominating sets in a simple undirected connected graph of order $n$ is in $Ω ({1.489}^{n})$ .

Proof.

By Lemma 1, each copy of the graph $G_{t}$ has $\frac{t^{3} + t^{2}}{2} - t$ minimal connected dominating sets. There are $k$ such graphs in $G_{t}^{k}$ , in addition to the vertex $s$ that connects them all. Every minimal connected dominating set must contain $s$ and at least one element from $N (s)$ in each $G_{t}$ . Therefore, the total number of minimal connected dominating sets in $G_{t}^{k}$ is $(\frac{t^{3} + t^{2}}{2} - t)^{k} = (\frac{t^{3} + t^{2}}{2} - t)^{\frac{n - 1}{2 t + 1}}$ . The claimed lower bound is achieved when $t = 4$ , which gives a total of $36^{\frac{n - 1}{9}} \in Ω ({1.489}^{n})$ . ∎

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

Figure 3: A plane drawing of

G_{3}^{3}

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 $n$ is in $Ω ({1.472}^{n})$ .

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 $G_{4}$ has 36 minimal connected dominating sets, each of which must contain elements of the set $X$ , which in turn is linked to the hub-vertex $s$ in $G_{4}^{k}$ .

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

[1] F. N. Abu-Khzam, A. E. Mouawad, and M. Liedloff, An exact algorithm for connected red-blue dominating set, J. Discrete Algorithms, 9 (2011), pp. 252–262.
[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.
[3] P. A. Golovach, P. Heggernes, and D. Kratsch, Enumerating minimal connected dominating sets in graphs of bounded chordality, Theor. Comput. Sci., 630 (2016), pp. 63–75.
[4] P. A. Golovach, P. Heggernes, D. Kratsch, and R. Saei, Enumeration of minimal connected dominating sets for chordal graphs, Discrete Applied Mathematics, 278 (2020), pp. 3–11. Eighth Workshop on Graph Classes, Optimization, and Width Parameters.
[5] D. Lokshtanov, M. Pilipczuk, and S. Saurabh, Below all subsets for minimal connected dominating set, SIAM J. Discret. Math., 32 (2018), pp. 2332–2345.
[6] I. B. Skjørten, Faster enumeration of minimal connected dominating sets in split graphs, Master’s thesis, The University of Bergen, 2017.

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

Authors

On the maximum number of minimal connected dominating sets in convex bipartite graphs

Enumerating Connected Dominating Sets

Online Connected Dominating Set Leasing

Upper paired domination versus upper domination

The growth rate over trees of any family of set defined by a monadic second order formula is semi-computable

Minimal Dominating Sets in a Tree: Counting, Enumeration, and Extremal Results

Minimal Roman Dominating Functions: Extensions and Enumeration

1 Introduction

2 Graphs with Large Minimal Connected Dominating Sets

Lemma 1.

Proof.

Theorem 1.

Proof.

Corollary 1.

3 Conclusion

References

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

Authors

Related Research

On the maximum number of minimal connected dominating sets in convex bipartite graphs

Enumerating Connected Dominating Sets

Online Connected Dominating Set Leasing

Upper paired domination versus upper domination

The growth rate over trees of any family of set defined by a monadic second order formula is semi-computable

Minimal Dominating Sets in a Tree: Counting, Enumeration, and Extremal Results

Minimal Roman Dominating Functions: Extensions and Enumeration

This week in AI

1 Introduction

2 Graphs with Large Minimal Connected Dominating Sets

Lemma 1.

Proof.

Theorem 1.

Proof.

Corollary 1.

3 Conclusion

References