Certified domination

Imagine that we are given a set $D$ of officials and a set $W$ of civils. For each civil $x \in W$, there must be an official $v \in D$ that can serve $x$, and whenever any such $v$ is serving $x$, there must also be another civil $w \in W$ that observes $v$, that is, $w$ may act as a kind of witness, to avoid any abuse from $v$. What is the minimum number of officials to guarantee such a service, assuming a given social network? In this paper, we introduce the concept of certified domination that perfectly models the aforementioned problem. Specifically, a dominating set $D$ of a graph $G=(V_G,E_G)$ is said to be certified if every vertex in $D$ has either zero or at least two neighbours in $V_G\setminus D$. The cardinality of a minimum certified dominating set in $G$ is called the certified domination number of $G$. Herein, we present the exact values of the certified domination number for some classes of graphs as well as provide some upper bounds on this parameter for arbitrary graphs. We then characterise a wide class of graphs with equal domination and certified domination numbers and characterise graphs with large values of certified domination numbers. Next, we examine the effects on the certified domination number when the graph is modified by deleting/adding an edge or a vertex. We also provide Nordhaus-Gaddum type inequalities for the certified domination number. Finally, we show that the (decision) certified domination problem is NP-complete.


Introduction
Imagine that we are given a set D of officials and a set W of civils. For each civil x ∈ W , there must be an official v ∈ D that can serve x, and whenever any such v is serving x, there must also be another civil w ∈ W that observes v, that is, w may act as a kind of witness, to avoid any abuse from v. What is the minimum number of officials to guarantee such a service, assuming a given social network?
The aforementioned problem motivates us introducing the concept of certified domination. Specifically, let D be a subsets of the vertex set of a graph G = (V G , E G ). We say that D dominates G (or is a dominating set of G) if each vertex in the set V G \ D has a neighbour in D. The cardinality of a minimum dominating set in G is called the domination number of G and denoted by γ(G), and any minimum dominating set of G is called a γ-set. A dominating set D of G is called certified if every vertex v ∈ D has either zero or at least two neighbours in V G \ D. The cardinality of a minimum certified dominating set in G is called the certified domination number of G and denoted by γ cer (G). A minimum certified dominating set of G is called a γ cer -set. Notice that, by the definition, V G is a certified dominating set of G, and certainly 1 ≤ γ cer (G) ≤ |V G |. Furthermore, one can observe that γ cer (G) = |V G | − 1.
There is a wealth of literature about domination and its variations in graphs; we refer to the excellent books of Haynes, Hedetniemi, and Slater [17,18]. The domination concept we introduce perfectly fits into that area where, for a given graph G, domination parameters are defined by imposing additional constraints on a dominating set D or its complement V G \ D. This area includes, to mention but a few, the multiple domination, the distance domination, or the global domination. In particular, the problem of certified domination is closely related to the problem of existence a DD 2 -pair in a graph, introduced by Henning and Rall in [19]. Recall, a set X ⊆ V G of vertices is 2-dominating in G if it is a dominating set of G and every vertex in V G \ X has at least two neighbours in X [14,15]. A DD 2 -pair of G is a pair (D, D 2 ) of disjoint sets of vertices of G such that D is a dominating set of G and D 2 is a 2-dominating set of G; a graph that has a DD 2 -pair is called a DD 2 -graph. One can observe that if G has a DD 2 -pair (D, D 2 ), then the set D is a certified dominating set. However, there are graphs G with γ cer (G) < |D| for any (D, D 2 )-pair in G (if any), see Fig. 1 for an illustration.  . . . Organization of the paper. In Section 2, we present the exact values of the certified domination number for some elementary classes of graphs. Some upper bounds on this new parameter for an arbitrary graph are presented in Section 3. Then, in Section 4 and Section 5, respectively, we characterise a wide class of graphs with equal domination and certified domination numbers and characterise graphs with large values of certified domination numbers. Next, in Section 6, we examine the effects on the certified domination number when the graph is modified by deleting/adding an edge or a vertex. Finally, Section 7 is devoted to Nordhaus-Gaddum type inequalities for the certified domination number, while in Section 8, we show that the (decision) certified domination problem is NP-complete. In addition, as a side result, in Section 9, we characterise a wider class of DD 2 -graphs, thus generalizing a result of [19].

Definitions and notation
For general graph theory terminology, we follow [10]. In particular, for a vertex v of , is the set of all vertices adjacent to v, and the cardinality of In general, for a subset X ⊆ V G of vertices, the (open) neighbourhood of X, denoted by N G (X), is defined to be v∈X N G (v), and the closed neighbourhood of X, denoted by N G [X], is the set N G (X) ∪ X. The minimum and maximum degree of a vertex in G is denoted by δ(G) and ∆(G), respectively. A vertex of degree |V G | − 1 is called a universal vertex of G. A vertex of degree one is called a leaf, and the only neighbour of a leaf is called its support vertex (or simply, its support). If a support vertex has at least two leaves as neighbours, we call it a strong support, otherwise it is a weak support. The set of leaves of G is denoted by L G . For a leaf v ∈ L G , its support vertex is denoted by s G (v), and for a weak support v, the unique leaf adjacent to v is denoted by l G (v). The set of weak supports of G is denoted by S 1 (G), while the set of strong supports of G is denoted by S 2 (G).

Elementary graph classes
We begin by presenting the exact values of the certified domination number for some elementary classes of graphs.
Observation 2.1. If P n is an n-vertex path, then Observation 2.2. If C n is an n-vertex cycle, n ≥ 3, then γ cer (C n ) = ⌈ n 3 ⌉.
Observation 2.3. If K n is an n-vertex complete graph, then Observation 2.4. If K m,n is a complete bipartite graph with 1 ≤ m ≤ n, then γ cer (K m,n ) = 1 if m = 1 and n > 1; 2 otherwise.
Observation 2.5. If W n is an n-vertex wheel, then γ cer (W n ) = 1.
In addition, we have the following two general observations on the certified domination number of a graph.
Observation 2.6. If G is a graph of order at least three, then γ cer (G) = 1 if and only if G has a universal vertex.
Observation 2.7. If G 1 , . . . , G k are the connected components of a graph G, then

Upper bounds on the certified domination number
In this section we focus on upper bounds on the certified domination number. We start with two simple observations and then present our main result of this section: an upper bound on γ cer (G) with respect to the domination number γ(G) and the number |S 1 (G)| of weak supports in G. Proof. Let s be a support vertex of G. If s were not in D, then l G (s) should be in D c . But then l G (s) would have only one neighbour in V G \ D c , and D c would not be a certified dominating set.
Observation 3.2. Let G be a graph of order n. If the strong supports of G are adjacent to k leaves in total, then γ cer (G) ≤ n − k. In particular, γ cer (G) ≤ n − 2 |S 2 (G)|.
Proof. Let L be the set of all leaf-neighbours of strong supports of G. Then |L| = k and the set V G \L is a certified dominating set of G.
Before we present our main result, let us introduce some useful terminology. Let D be a dominating set of a graph G. An element of D that has all neighbours in D is said to be shadowed with respect to D (shortly shadowed ), an element of D that has exactly one neighbour in V G \ D is said to be half-shadowed with respect to D (shortly half-shadowed ), while an element of D having at least two neighbours in V G \ D is said to be illuminated with respect to D (shortly illuminated ). It is easy to observe that if D is a minimum dominating set of a graph with no isolated vertices, then D has no shadowed element, and if D is a certified dominating set, then D has no half-shadowed element.
Proof. If G is a graph of order at most two, then the inequality is obvious. Thus assume that G has at least three vertices. Let D be a γ-set of G that minimizes the number of half-shadowed vertices and such that D does not contain any leaf of G. (Notice that such D always exists as G is connected and |V G | ≥ 3.) Let D hs ⊆ D be the set of all half-shadowed vertices of D. If D hs = ∅, then γ cer (G) = γ(G) ≤ γ(G) + |S 1 (G)|. Thus assume that D hs = ∅.
The inequality deg G (v) ≥ 2 follows from the choice of D, that is, from the assumption that D ∩ L G = ∅. To argue the second property, suppose on the contrary that v is a strong support. Again, since L G ∩ D = ∅ and v has at least two leaf-neighbours, v would not be half-shadowed, a contradiction.
Next we show that all half-shadowed vertices are weak supports. Suppose on the contrary that there is a half-shadowed vertex v ∈ D hs \ S 1 (G) and let u be the unique neighbour Since v is neither a weak nor strong support (by assumption and Claim 1, respectively), it implies that u is not a leaf. Furthermore, we have the following claim.
Otherwise the set D \ {v} would be a smaller (than D) dominating set of G.
On the other hand, as regards all (but u) neighbours of v that are in D, we have the following claim.
Otherwise the set D \ {w} would be a smaller (than D) dominating set of G.
Consequently, keeping in mind the fact that none of neighbours of v is a leaf (see Claim 1), by combining Claims 2 and 3, we conclude that the set (D \ {v}) ∪ {u} would be a dominating set with a smaller number of half-shadowed vertices, a contradiction. Therefore, the set D hs of half-shadowed vertices consists of weak supports of G only. Observe now that adding to D all leaves adjacent to half-shadowed weak supports results in a dominating set D ′ of G with no half-shadowed vertices, that is, D ′ is a certified dominating set of G. Therefore γ cer (G) From Observation 2.7 and Theorem 3.3, we immediately obtain the following corollary.
Finally, we have the following corollary. We emphasize that the above upper bounds are sharp: the bound γ cer (G) ≤ γ(G) + |S 1 (G)| in terms of |S 1 (G)|, while the bound γ cer (G) ≤ 2γ(G) in terms of γ(G), and their sharpnesses are established by coronas of graphs which we shall discuss in Section 5 (see Remark on page 7).

Graphs with γ cer = γ
We continue our study on the certified domination number by focusing now on the class of graphs with γ cer = γ. When trying to characterise this class, one may expect that the main problem lies in leaves of a graph. In fact, from the inequalities γ(G) ≤ γ cer (G) ≤ γ(G)+|S 1 (G)| (see Corollary 3.4), we immediately have the first two results of this section.  The above two corollaries also follow from the next more general lemma.
. Of all such sets, choose one, say D ′ , that does not contain any leaf of G (such D ′ exists in every connected graph of order at least three) and minimizes the number of its half-shadowed vertices. We claim that such D ′ is a certified dominating set of G (and therefore γ(G) = |D ′ | = γ cer (G)). Suppose, on the contrary, that some element and v ∈ S 1 (G) (as every element of S 1 (G) is illuminated by the adjacent leaf and, by the assumption, by at least one non-leaf). Finally, since is a minimum dominating set of G and it has less half-shadowed vertices than D ′ , a final contradiction which proves that γ(G) = γ cer (G).
Observe that if G = K n , then γ cer (G) = n = γ(G). Next, if G = lK 2 , then γ cer (G) = 2l = l = γ(G). In the latter case, S 1 (G) = V G = L G and G has no minimum dominating set D of G such that N G (s) \ L G ⊆ D for every s ∈ S 1 (G). Therefore, taking into account Observation 2.7 and Lemma 4.3, we obtain the following corollary for graphs which are not necessarily connected.
Furthermore, we have the following relation between graphs each of which has a unique minimum dominating set and those for which γ cer and γ are equal.

Graphs with large values of γ cer
As we have already observed, for any graph G of order n, γ cer (G) ≤ n, γ cer (G) = n − 1, and there are graphs G with γ cer (G) = n, for example, the complement of a complete graph K n or a 4-vertex path P 4 . Thus it is natural to try to characterise all graphs with γ cer = n and γ cer = n − 2, respectively, which is successfully carried out in this section.
In particular, we prove that γ cer (G) = n if and only if G is the complement of a complete graph, the corona of a graph, or the union of both of them. Recall, the corona product (or simply, the corona) of two graphs H and F is the graph G = H • F resulting from the disjoint union of H and |V H | copies of F in which the i-th vertex of H is joined to all vertices of the i-th copy of F . If F is a 1-vertex graph, F = K 1 , then the corona H • K 1 is simply called the corona of H.
Lemma 5.1. Let G be a connected graph of order n. If G is the corona of some graph, then γ cer (G) = n.
Proof. Let D c be a smallest certified dominating set of G. It suffices to prove that D c = V G . This is obvious if n = 2. Thus assume n > 2. In this case, since G is the corona of some graph, every vertex of G either is a leaf of G or is adjacent to exactly one leaf of G. From this and from Observation 3.1 it follows that V G \ L G ⊆ D c . Moreover, every leaf l of G also belongs to D c (as otherwise its only neighbour s G (l) would be half-shadowed).
Proof. The statement is obvious for connected graphs of order at most 4. Thus assume that G is a connected graph of order n ≥ 5 and γ cer (G) = n. Now, since γ(G) ≤ n/2 for every graph with no isolated vertex, so by Corollary 3.5 we have γ cer (G) ≤ 2γ(G) ≤ n = γ cer (G). Thus γ(G) = n/2 and so G is the corona of some graph (as it was proved in [16]).
From the above lemmas, we immediately conclude with the following theorem. Remark. We incidentally observe that the above result implies the sharpness of the upper bound γ cer (G) ≤ γ(G) + |S 1 (G)| (Corollary 3.4) in terms of |S 1 (G)| as well as the sharpness of the upper bound γ cer (G) ≤ 2γ(G) (Corollary 3.5) in terms of γ(G), since for the corona G of any graph, we have |S 1 (G)| = γ(G) and γ cer (G) = 2γ(G).

Graphs with γ cer = n − 2
A diadem graph of a graph H is a graph obtained from the corona H • K 1 by adding a new vertex, say v, and joining v to one of support vertices of H • K 1 (see Fig. 2). Proof. Let s be the unique strong support of G, and let l 1 , l 2 be the two leaves of G adjacent to s in G. It is obvious that V G \ {l 1 , l 2 } is a certified dominating set of G. Let D c be a smallest certified dominating set of G. Then V G \ L G ⊆ D c (by Observation 3.1) and {l 1 , l 2 } ∩ D c = ∅. Moreover, every leaf l different from l 1 and l 2 belongs to D c (otherwise s G (l) would be half-shadowed). Consequently D c = V G \ {l 1 , l 2 } and therefore γ cer (G) = n − 2.
Lemma 5.5. Let G be a connected graph of order n. If γ cer (G) = n − 2, then G = C 3 , a diadem graph (of a connected graph).
By way of contradiction, suppose that the claim is false. Let G be a smallest counterexample, say of order n (n ≥ 5), such that γ cer (G) = n−2 and G is not a diadem graph. Let D c be a γ cer -set of G, and let v and u be the only elements of In addition, the set V G \ N G [{v, u}] is nonempty, as otherwise {v, u} would be a certified dominating set of G and we would have n − 2 = γ cer (G) ≤ |{v, u}| = 2, which is impossible.
Let G ′ denote the subgraph G −N G [{v, u}] of G. From the assumption γ cer (G) = n−2 it easily follows that γ cer (G ′ ) = |V G ′ |. Thus, by Theorem 5.3, every connected component of G ′ is an isolated vertex or the corona of a graph.
Let H be a connected component of G ′ . From the fact that D c = V G \ {v, u} is a minimum certified dominating set of G it follows that at least one vertex of H is not adjacent to any vertex belonging to N G [{v, u}] \ {v, u} as otherwise D c \ V H would be a certified dominating set of G, which is impossible as γ cer (G) ≤ |D c \V H | < |D c | = γ cer (G). From this we conclude that G ′ has no isolated vertex. Consequently, every connected component of G ′ is the corona of a graph.
We now claim that K 2 is not a connected component of G ′ . Suppose on the contrary that K 2 on vertices a and b is a connected component of G ′ . Then one of the vertices a and b is a leaf in G and the latter one is adjacent to a vertex in N G [{v, u}] \ {v, u}, say a ∈ L G and b is adjacent to a vertex w ∈ N G [{v, u}]\{v, u}. Let G denote the graph G−{a, b} (of order n − 2). For this graph either γ cer ( G) < n − 4, or γ cer ( G) = n − 4, or γ cer ( G) > n − 4. Assume first that γ cer ( G) < n − 4. Let D c be a smallest certified dominating set of G.
is a certified dominating set of G and γ cer (G) ≤ | D c ∪ {a, b}| = γ cer ( G) + 2 < n − 2, a contradiction. Assume now that γ cer ( G) > n − 4. Then γ cer ( G) = n − 2 = |V G | and, by Theorem 5.3, G is the corona of a graph. But this is impossible as no vertex of N G [{v, u}] \ {v, u} is a leaf or a neighbour of exactly one leaf. Finally, assume that γ cer ( G) = n − 4 = |V G | − 2. In this case the choice of G implies that G is the diadem graph in which v and u are leaves and w is their only common neighbour. Now, it is obvious that the graph G (obtained from G by the addition of the vertices a and b, and the edges ab and bw) is a diadem graph, a contradiction. Now, to complete the proof, it suffices to show that this smallest counterexample is not a counterexample, that is, it suffices to show that G is a diadem graph. It is enough to prove that: (1) is a certified dominating set of G and |D| < |D c |, a contradiction.
(3) Suppose on the contrary that vu ∈ E G , and consider the graph G ′′ = G−vu of order n, in which, by (2), v and u are leaves, and they have exactly one common neighbour, say w. In this graph we have either γ cer (G ′′ ) > n − 2 (and therefore γ cer (G ′′ ) = n), or γ cer (G ′′ ) = n − 2, or γ cer (G ′′ ) < n − 2. Assume first that γ cer (G ′′ ) = n. Then, by Theorem 5.3, G ′′ is the corona of a graph, but this is impossible as leaves v and v share the same neighbour w. Assume now that γ cer (G ′′ ) = n−2. Then, by the choice of G, G ′′ is a diadem graph. Let L be the set of leaves of G ′′ within the distance at most 3 from v (and u). Then D = (D c \ (L ∪ {w})) ∪ {v} is a certified dominating set of G and |D| < |D c |, a contradiction. Finally, assume that γ cer (G ′′ ) < n − 2. Let D ′′ c be a smallest certified dominating of G ′′ . Since w is a strong support of G ′′ , w ∈ D ′′ c by Observation 3.1, and v, u / ∈ D ′′ c by minimality of D ′′ c . But then, D ′′ c is also a certified dominating set of G and so γ cer (G) < n − 2, a final contradiction. Theorem 5.6. Let G be a graph of order n ≥ 3. Then γ cer (G) = n − 2 if and only if G is C 3 , C 4 , or a diadem graph, or G is one of these three graphs with possible number of isolated vertices, or G is the union of one of these three graphs with the corona of some graph, with possible number of isolated vertices.

Influence of deleting/adding edge/vertex
In this section, following [4,5,7,12], to mention but a recent few, we examine the effects on the certified domination number when the graph is modified by deleting/adding an edge or a vertex. We observe that deleting an edge or a vertex may arbitrarily increase the certified domination number. For example, for the graph G i of order 2i + 4 illustrated in Fig. 3(a) we have γ cer (G i ) = i + 1 and γ cer (G i − e) = 2i + 4. To argue a similar influence of deleting a vertex, consider a wheel graph W n with the hub v. We have γ cer (W n ) = 1 and γ cer ( Fig. 3. Adding or deleting an edge may arbitrarily increase the certified domination number. Adding an edge to a graph may also arbitrarily increase the certified domination number. Namely, consider the disconnected graph H i of order 2i+4 illustrated in Fig. 3(b). We have γ cer (H i ) = i+2 and γ cer (H i +e) = 2i+4. However, adding an edge to a connected graph does not increase the certified domination number, that is, γ cer (G + e) ≤ γ cer (G) for any connected graph G. To argue this property, we use the following lemma. Proof. (a) Consider a shadowed vertex v ∈ D c . Suppose on the contrary that v is neither a weak support nor a leaf in G. By minimality of D c , there are no shadowed strong supports in D c , in particular, v is not a strong support, and thus all neighbours of v are of degree at least two. Let X ⊆ D c be a maximal subset of shadowed vertices such that (i) v ∈ X, (ii) the induced subgraph G[X] is connected, and (iii) none of elements of X is an illuminated vertex or a shadowed weak support. Next, for a vertex x ∈ X, define the set Observe that by minimality of D c , each vertex x ∈ X is a non-support vertex, and by the choice of X, and every element in B G (X) is either an illuminated vertex or a shadowed weak support. Let D X be a γ cer -set of G[X] and let D X = X \ D X . Let L 3 be the set of shadowed leaves within the distance 2 from D X . Then the set D = D c \ (D X ∪ L 3 ) is a certified dominating set of G and |D| < |D c |, a contradiction.
(b) A non-leaf neighbour of a shadowed weak support s ∈ D c is either illuminated or shadowed. If s is shadowed, then, since it is not a leaf, it must be a weak support by (a). Theorem 6.2. If G is a connected graph of order n ≥ 2, then γ cer (G + e) ≤ γ cer (G).
Proof. One can verify the validity of the theorem for graphs of order at most n ≤ 4. So assume n ≥ 5 and let D c be a γ cer -set of G.
Let e = vw, v, w ∈ V G , be the added edge to G. If both v, w ∈ D c , then D c is also a certified dominating set of the graph G + e. Similarly, if either both v, w / ∈ D c , or v / ∈ D c and w ∈ D c is illuminated, or v ∈ D c is illuminated and w / ∈ D c , then D c is a certified dominating set of G + e as well. Therefore, in all aforementioned cases, we have γ cer (G + e) ≤ |D| = γ cer (G) as required.
Without loss of generality assume now that v / ∈ D c and w ∈ D c is shadowed (the case w / ∈ D c and v ∈ D c is shadowed can be analysed in a similar way). By Lemma 6.1(a), w is either a weak support or a leaf of G.
Case 1: w is a weak support of G. Then the set D = D c \{l G (w)} is a certified dominating set of G + e, and thus, γ cer (G + e) ≤ |D| < |D c | = γ cer (G).
Case 2: w is a leaf of G. By the choice of D c , it follows that the support vertex s G (w) is weak and shadowed. Therefore, by Lemma 6.1(b), every non-leaf neighbour of the weak support s G (w) in G is either an illuminated vertex or a shadowed weak support of G. Let L be the set of shadowed leaves within the distance 2 from s G (w) in G. Then, the set D = D c \ (L ∪ {s G (w)}) is a certified dominating set in G + e, and hence γ cer (G + e) ≤ |D| ≤ |D c | − 1 < γ cer (G). Fig. 4. Graph G i has 2i + 1 vertices, and γ cer (G i ) = i, while γ cer (G i + v) = 2i + 2.
As regards adding a vertex, we claim that it may arbitrarily increase the certified domination number which is not the case as in the model of classic domination. Indeed, for the graph G i of order 2i+1 depicted in Fig. 4, we have γ cer (G i ) = i, while γ cer (G i +v) = 2i + 2. However, bearing in mind Corollary 4.1, one can expect that the clue of the above construction lies in adding a leaf. Indeed, this is the case since one can prove that adding a non-leaf vertex does not effect the certified domination number significantly. Namely, we have the following theorem. If we add a non-leaf vertex v to a graph G, then γ cer Proof. Let D c be a γ cer -set of a graph G and let v be a new added vertex.
Case 1: deg G+v (v) = 2. Let u and w be the two neighbours of v in G + v. If either v, w / ∈ D c or both v, w ∈ D c , then the set D c ∪ {v} is a certified dominating set of G + v, and thus γ cer (G + v) ≤ γ cer (G) + 1. Otherwise, without loss of generality, we consider two subcases. Subcase 1.a: u / ∈ D c , w ∈ D c , and w is illuminated. Then the set D c remains a certified dominating set of G + v, and in this case, γ cer (G + v) ≤ γ cer (G) holds. Subcase 1.b: u / ∈ D c , w ∈ D c , and w is shadowed. If the vertex w constitutes a 1-vertex component of G, then the set (D c \ {w}) ∪ {v} is a certified dominating set in G + v, thus getting γ cer (G + v) ≤ γ cer (G) + 1. Otherwise, by Lemma 6.1(a), w is either a weak support or a leaf of G. Now, similarly as in the proof of Theorem 6.2, we consider two subcases. Case 2: deg G+v (v) ≥ 3. Then, when adding v to G, we first add only two edges, thus obtaining a temporary graph G ′ , where deg G ′ (v) = 2. Now, taking into account Case 1, we conclude that γ cer (G ′ ) ≤ γ cer (G) + 1. Next, when adding all the remaining edges to G ′ , sequentially, to obtain the final graph G + v, we apply Theorem 6.2, sequentially, for each of added edge, thus getting γ cer (G + v) ≤ γ cer (G ′ ) ≤ γ cer (G) + 1 as required.

Nordhaus-Gaddum type results
Following the precursory paper of Nordhaus and Gaddum [28], the literature has became abundant in inequalities of a similar type for many graph invariants, see a recent survey by Aouchiche and Hansen [1]. In particular, the following result is known for the domination number. Further sharpening of bounds was done for the case when, for example, both G and G are connected [25] or for graphs with specified minimum degree [11], to mention but a few. In particular, the following theorem was proved in [22]. Theorem 7.2. [22] If a graph G is a graph of order n and neither G nor G has an isolated vertex, that is 1 ≤ δ(G) ≤ n − 2, then Moreover, if n = 9, the bound is attained if and only if {γ(G), γ(G)} = { n 2 , 2}. In this section we provide some Nordhaus-Gaddum type inequalities for the certified domination number. First, taking into account Corollary 4.1, Theorem 7.1 and Theorem 7.2, we obtain the following corollary. By enumerating all graphs of order at most 4, we obtain the following observation. Proof. From the assumption min{δ(G), δ(G)} = 0 it follows that max{∆(G), ∆(G)} = n − 1 and, therefore, γ cer (G) = 1 or γ cer (G) = 1. Now, since γ cer (G) ≤ n and γ cer (G) ≤ n, we get γ cer (G) + γ cer (G) ≤ n + 1 and γ cer (G) γ cer (G) ≤ n. Assume now that min{δ(G), δ(G)} = 0, say δ(G) = 0. Then ∆(G) = n − 1, and so γ cer (G) = 1. Now, since γ cer (G) = 1, it follows from each of the equalities γ cer (G) + γ cer (G) = n + 1 and γ cer (G) γ cer (G) = n that γ cer (G) = n. Finally, since δ(G) = 0 and γ cer (G) = n, we conclude from Theorem 5.3 that G is the complement of K n or the union of the corona of some graph and a positive number of isolated vertices. This proves the implications a) ⇒ c) and b) ⇒ c). Opposite implications are straightforward.
In addition, each of the above upper bounds is attainable, and the following statements are equivalent: a) γ cer (G) + γ cer (G) = n + 2; b) γ cer (G) γ cer (G) = 2n; c) G or G is the corona of some graph.
When introducing a new model of domination, no discussion would be complete without mentioning its complexity status. Define the (decision) certified domination problem as follows.
The certified domination problem Instance: A graph G and an integer k ≥ 1. Question: Does there exist a certified dominating set of G of cardinality at most k?
As one can expect, the certified domination problem is NP-complete. Specifically, the certified domination problem remains NP-complete even if we restrict ourselves to bipartite planar subcubic graphs. The proof immediately follows from Corollary 4.1 and the fact that the (original) domination problem is NP-complete in bipartite planar subcubic graphs with no leaves [24,29].
Theorem 8.1. The certified domination problem is NP-complete in bipartite planar subcubic graphs with no leaves.
Following several algorithmic results for the domination problem and its variations [6,17,23], one can focus on designing effective (exponential time) algorithms, faster (polynomial time) algorithms for some restricted graph classes, or approximation algorithms for the variant of certified domination.

Certified domination and DD 2 -graphs
Here, we briefly continue our discussion from the introductory section on the relation between the concepts of certified domination and DD 2 -graphs. In [19], the following theorem was established.
Theorem 9.1. [19] A graph of a minimum degree at least two is a DD 2 -graph.
Since any minimal dominating set has no shadowed vertices, one can observe that if a graph G has a minimal dominating set D being also certified then its complement V G \ D is a 2-dominating set of G. Therefore, taking into account Corollaries 4.1 and 4.5, we conclude with the following theorem which generalizes Theorem 9.1. b) A graph of order at least two having a unique dominating set is a DD 2 -graph.
Moreover, any such graph in a) or b) has a (D, D 2 )-pair such that |D| = γ(G).

Concluding remarks
Since over the years researchers have published thousands of papers on the topic of domination in graphs, our paper cannot claim the right to cover the new model even partially, it should only be thought of as a very beginning, a small contribution to. In this section, we present two exemplary open problems that we find interesting and which research on we feel worth of being continued.
Stable and critical graphs. It is natural to characterise the class of critical graphs where the certified domination number increases on the removal of any edge/vertex as well as the class of stable graphs where the certified domination number remains unchanged on the removal of any edge/vertex. We point out that by Corollary 4.1, the class of critical (resp. stable) (with respect to the certified domination number) graphs with minimum degree δ ≥ 3 is the same as the class of critical (resp. stable) graphs with respect to domination number, see for example [3,13,27]. Therefore, we are left with characterising critical (resp. stable) graphs with minimum degree δ ≤ 2. This is an open problem.
Trees with γ cer = γ. The problem of constructive characterisations of trees with equal domination parameters has received attention in the literature, see for example [9,20,26], to mention but a few recent. Following this concept, we leave as an open problem to provide a constructive characterisation of (γ, γ cer )-trees, that is, the class of trees with γ cer = γ.