THE TOTAL CO-INDEPENDENT DOMINATION NUMBER OF SOME GRAPH OPERATIONS

. A set D of vertices of a graph G is a total dominating set if every vertex of G is adjacent to at least one vertex of D . The total domi- nating set D is called a total co-independent dominating set if the subgraph induced by V ( G ) − D is edgeless. The minimum cardinality among all total co-independent dominating sets of G is the total co-independent domination number of G . In this article we study the total co-independent domination number of the join, strong, lexicographic, direct and rooted products of graphs.


Introduction
Throughout this article we consider simple graphs G = (V (G), E(G)) of order n and size m. That is, graphs that are finite, undirected, and without loops or multiple edges. Given a vertex v of G, N (v) represents the open neighborhood of v, i.e., the set of all neighbors of v in G, and the degree of v is δ(v) = |N (v)|. If δ(v) = n − 1, then we say that v is a universal vertex of G. The minimum and maximum degrees of G are denoted by δ(G) and ∆(G), respectively. By G [D] we denote the induced subgraph of G on D ⊆ V (G). By the union of two graphs G ∪ H we mean the disjoint union. In particular, kG = ∪ k i=1 G is a disjoint union of k copies of a graph G. We use the notation [k] for the set of integer numbers {1, . . . , k}.
Given a graph G, a set D ⊆ V (G) is a total dominating set of G if every vertex in V (G) is adjacent to at least one vertex in D. The total domination number of G is the minimum cardinality among all total dominating sets of G and is denoted by γ t (G). A γ t (G)-set is a total dominating set of cardinality γ t (G). For more information on total domination we suggest the relatively recent survey [15] and the book [16]. A set of vertices is independent if it induces an edgeless graph. The independence number of G is the cardinality of a maximum independent set of G and is denoted by α(G). An independent set of cardinality α(G) is called an α(G)-set. A set S of vertices of G is a vertex cover if every edge of G is incident in at least one vertex of S. The vertex cover number of G, denoted by β(G), is the smallest cardinality among all vertex covers of G. We refer to a β(G)-set in G as a vertex cover of cardinality β(G). The following well-known result, due to Gallai [12], states the relationship between the independence number and the vertex cover number of a graph. Theorem 1.1 (Gallai, 1959 [12]). For any graph G of order n, β(G) + α(G) = n.
The theory of domination and independence in graphs has attracted the attention of many researchers since several years ago. The number of works, results and open problems in this area span a wide range of directions, from highly theoretical aspects and various practical applications, to a large number of connections with other topics in graph theory itself and in some external areas. One can easily notice these facts, by simply making some specialized searches in well-known databases like MathSciNet, Scopus or JCR. The idea of studying a variant of a dominating set whose complement is independent has previously been explored, for instance, in the Ph.D. thesis [19], although it was probably introduced earlier.
In recent years, a reborn interest has arisen in research concerning connections between domination and independence in graphs. One of the ideas behind this interest comes from a Roman domination structure whose "complement" has independent properties. Particular and remarkable cases are observed in [1,8] for co-independent Roman domination, in [2] for co-independent double Roman domination, and in [7,18] for co-independent total Roman domination. Other similar parameters not related to Roman domination are [3,4,11]. It is then our goal here to continue making some contributions to this topic which concerns dominating sets whose complements form an independent set. We remark that some of the articles referenced above use the term "outer-independence" instead of "co-independence".
A total dominating set D of a graph G without isolated vertices is called a total co-independent dominating set (or TC-ID set for short) if the set of vertices V (G) − D is independent. The total co-independent domination number of G is the minimum cardinality among all TC-ID sets of G and is denoted by γ t,coi (G). A TC-ID set of cardinality γ t,coi (G) is a γ t,coi (G)-set.
The total co-independent domination number of a graph G has been introduced in [23], where a few of its combinatorial properties were considered. Among them, a couple of almost trivial bounds in terms of α(G) and the order of G were proved for γ t,coi (G). The following result is an example of this.
Proof. By Lemma 2.1 and Theorem 1.2 we obtain the inequality γ t,coi are TC-ID sets of G ∨ H whenever both G and H, respectively, contain at least one edge. In this case γ t,coi (G ∨ H) ≤ min{|D G |, |D H |} = n G + n H − max{α(G), α(H)}. Therefore the equality follows when m G > 0 and m H > 0. Suppose next that m G > 0 and m H = 0, which means that α(H) = n H and , α(H)} and equality holds again. So, let α(G) < n H = t. If δ(G) > 0, then D H = V (G) is a TC-ID set because V (G) contains no isolated vertex. In this case γ t,coi (G ∨ H) ≤ |D H | = n G = n G + n H − t = n G + n H − max{α(G), α(H)} and equality holds again. If δ(G) = 0, then D H = V (G) is not a TC-ID set because D H is not a total dominating set of G ∨ H. Notice that V (H) is the unique α(G ∨ H)set because α(G) < n H and every independent set of G ∨ H is contained either in Finally, let m H = m G = 0, which means that G ∼ = K s and H ∼ = K t . Again D G and D H are not TC-ID sets of G ∨ H because V (H) and V (G), respectively, are not total dominating sets of G ∨ H. As before, the inequality γ t,coi Let G and H be two graphs. The lexicographic product of G and H is the graph presents a connection between the lexicographic product and the join of graphs.
In order to obtain the total co-independent domination number of the lexicographic product of two graphs, we need the following result.

Theorem 2.3 ([13]). For any two graphs G and H, α(G • H) = α(G)α(H).
It is straightforward to observe that G • H is connected whenever G K 1 is connected. Moreover, if G is not connected, then one can treat every component of G • H separately with respect to γ t,coi (G • H). Hence, we can assume that G is connected.
Let G be a graph without isolated vertices. By I(G) we denote the set of all maximal independent sets of G. By a maximal independent set we mean an independent set which is not contained in any other independent set (notice that any maximum independent set is maximal, but the contrary is not always true).
There are exactly two maximum independent sets in I(G), namely A G,1 = {v 2i : i ∈ [k]} and A G,2 = {v 2i−1 : i ∈ [k]}. Without loss of generality we may assume that A G = A G,2 because A G,1 and A G,2 are symmetric. It is easy to see that one possibility is A * G = {v 4i : i ∈ [k/2]} if k is even, and The equality follows by Theorem 1.2 and Theorem 2.3.
Let now H ∼ = K t and let first . Hence, to obtain a TC-ID set we need to add some vertices.
We will see that every vertex g ∈ A * G is adjacent to some vertex g ∈ D G , which is an isolated vertex in the subgraph of G induced by D G . If this does not hold, then we obtain a contradiction with D being a γ t,coi Combining both inequalities yields the stated equality.
Notice that I(G) contains all maximal independent sets, which are not necessarily α(G)-sets. We are not aware of any example where a maximal independent set (that is not an α(G)-set) would yield a better result in Theorem 2.4 than an α(G)-set.
The strong product of the graphs G and H is the graph Also, G H has isolated vertices whenever both G and H have isolated vertices. By H − we denote the graph obtained from H by deleting all isolated vertices. The upper bound γ t (G H) ≤ γ t (G)γ t (H) follows from a result in [20]. The independence number α(G H) is the basis for the so-called Shannon capacity; see [ Together with all the components of G H isomorphic to G, we obtain the desired result.

The direct product of graphs
The direct product of two graphs G and H is the graph G × H, with vertex set , h ∈ V (H)}, and two vertices (g, h) and (g , h ) are adjacent in G × H if and only if gg ∈ E(G) and hh ∈ E(H). The direct product can be considered as a subgraph of the strong product. It has the special property that every edge from G × H projects to edges in both factors G and H, and it is therefore often considered as the most natural graph product. On the other hand, this brings several problems. Even connectivity is not trivial among direct products. Indeed, G × H is a connected graph if and only if both G and H are connected and at least one of G and H is non bipartite. Moreover, if both G and H are bipartite, then G × H has exactly two components (see [25] and also [14]). In addition, and important in our case, G × H has isolated vertices if and only if at least one of G and H has isolated vertices. G-and H-layers are defined as in the lexicographic product. However, one must note that any layer induces a graph without edges. This already implies the lower bound A remarkable upper bound for α(G × H) can be found in [24]. Some bounds and exact results on γ t (G × H) can be found in [9,10,20,21].
It is easy to see that K p,q × K r,s ∼ = K pr,qs ∪ K ps,qr . Since K pr,qs ∼ = K pr ∨ K qs and K ps,qr ∼ = K ps ∨ K qr , we can use Theorem 2.2 to obtain γ t,coi (K p,q × K r,s ) = min{pr, qs} + min{ps, qr} + 2. (3.1) We next present an upper bound for γ t,coi (G × H) from the independent set perspective. Let G be a graph without isolated vertices. Recall that I(G) is the set of all maximal independent sets of G, that D G = V (G) − A G is a dominating set of G for any A G ∈ I(G), and that A * G is a minimum subset of A G such that A * G dominates the set (denoted by D * G ) of all isolated vertices of G[D G ].
Proof. Let A G be any maximal independent set of a graph G and let T H be a γ t (H)-set. We will show that S = ( Next we show that S is a total dominating set of G × H. Let (u, v) be an arbitrary vertex of G × H. If u ∈ A G , then there exists g ∈ D G which is adjacent to u in G because D G dominates G. There also exists h ∈ V (H) which is adjacent to v in H because H has no isolated vertices and (g, h) ∈ S is adjacent to (u, v). Again we are not aware of any example where a maximal independent set (that is not an α(G)-set) would yield a better result in Theorem 3.1 than an α(G)-set.
The bound of Theorem 3.1 performs rather well. We show this by the results that follow until the end of this section. However, we do not obtain equality in all cases. The smallest example is probably C 5 × K 2 ∼ = C 10 . We have γ t,coi (C 10 ) = 7, but the bound from Theorem 3.1 gives 8. Moreover, this is the smallest member of the family C 6k−1 × K 2 ∼ = C 12k−2 , for an arbitrary positive integer k, for which γ t,coi (C 12k−2 ) = 8k − 1, but the upper bound from Theorem 3.1 gives 8k. Another example is C 5 × C 5 , where it is not hard to check that is a TC-ID set of C 5 × C 5 , by taking g 0 g 1 g 2 g 3 g 4 g 0 as the first cycle C 5 and h 0 h 1 h 2 h 3 h 4 h 0 as the second C 5 . With this we have γ t,coi (C 5 × C 5 ) ≤ 17, but Theorem 3.1 gives 18.
We can use the example K p,q × K r,s to show that the bound of Theorem 3.1 is sharp. By some straightforward computations, one obtains from Theorem 3.1 that γ t,coi (K p,q × K r,s ) ≤ min{(r + s) min{p, q} + 2, (p + q) min{r, s} + 2}. With an additional analysis of the relationship ≥ between p and q, as well as between r and s, we obtain the same result as in (3.1).
We next study the total co-independent domination number of several examples of direct products. They all show that the bound of Theorem 3.1 is tight.
Let V r and V t be the bipartite sets of K r,t of cardinality r and t, respectively. Let S be a γ t,coi (K r,t × K n )-set and let S r = S ∩ (V r × V (K n )) and The set S t is independent since V t is independent, and therefore S r = ∅ because S is a total dominating set. If S r = {(g, h)}, then the vertices from V t × {h} have no neighbor in S, a contradiction. Hence S r contains at least two vertices. This means that we have γ t,coi (K r,t × K n ) ≥ nt + 2 in this case. We can ignore the symmetric condition V r × V (K n ) ⊆ S r because we obtain at least nr + 2 ≥ nt + 2 vertices in |S|, a contradiction with S being a γ t,coi (K r,t × K n )-set when r > t.
Now we may assume that V t × V (K n ) = S t and V r × V (K n ) = S r . Since any two vertices (g, h), (g , h ) such that g ∈ V r , g ∈ V t and h = h are adjacent, it must happen, for the complement of S to be independent, that V r × (V (K n ) − {h}) ⊆ S r and V t × (V (K n ) − {h}) ⊆ S t for some h ∈ V (K n ). Consequently, |S| = |S r | + |S t | ≥ (r + t)(n − 1) = rn + tn − r − t. Since r ≥ t ≥ 2 and n ≥ 3, it happens that rn − r − t ≥ rn − 2r = r(n − 2) ≥ r ≥ 2. Thus, again we have γ t,coi (K r,t × K n ) = |S| ≥ tn + 2.

Proposition 3.4. For any integers r, t with r, t ≥ 4,
Proof. Let C r = g 0 g 1 . . . g r−1 g 0 (from now on, operations with the subindexes of such vertices are done modulo r) and let V ( Suppose first that there exists an edge between two vertices of p Cr (A), that is, for instance, and K gi+3 t must be in S because γ t (K t ) = 2. We conclude that among any four consecutive and K gi t at least two of them are completely contained in S and in addition one of the other two contains at least γ t (K t ) = 2 vertices in S. Since this is better than in the previous paragraph, we can ignore that option. This means that for every i ∈ {0, . . . , r−1}, |S∩({g i−3 , g i−2 , g i−1 , g i }×V (K t ))| ≥ 2t + 2. Therefore, Notice that in the above argument we can have also K gi+4 t ⊆ S for S to be coindependent. In this case we have only 2t in S ∩ {g i−3 , g i−2 , g i−1 , g i }, but on the other hand at least 3t vertices in S ∩ {g i−4 , g i−3 , g i−2 , g i−1 }. Because t ≥ 4 this approach gives more vertices in S and we can ignore it. Next we proceed by analyzing four different situations.
Case 1: r ≡ 0 (mod 4). Clearly, A Cr = {g 2i−1 : i ∈ [r/2]} is in I(C r ) and, in this case, A * Cr = {g 4i−1 : i ∈ [r/4]} is one option. By Theorem 3.1, we obtain γ t,coi (C r × K t ) ≤ min{ rt 2 + 2r 4 , (t − 1)r + 0} = r 2 (t + 1), where the last equality occurs since t ≥ 3. The other inequality comes from (3.2), and we have the desired equality in this case.  For the lower bound, notice that the comments before (3.2) allow us to claim that there exist at least four consecutive K t -layers, all four of them having nonempty intersection with S, because r ≡ 1 (mod 4). Furthermore, among these four layers, at least three of them are completely contained in S. One of them is treated separately in the sum of (3.2), and we get , (t − 1)r + 0} = r 2 (t + 1) + 1, where the last equality is due to the fact that t ≥ 4 and r ≥ 3.
For the lower bound, we use a similar argument as in Case 2. Now, there exist at least five consecutive K t -layers, all of them with nonempty intersection with S, because r ≡ 2 (mod 4). Clearly, at least three of them are completely contained in S. (Notice that one layer with empty intersection with S is possible among them, however this yields that the remaining four are completely contained in S, which is not optimal as t ≥ 4.) We deal with two of them separately in the sum of (3.2), and obtain For the lower bound, there exist at least two consecutive K t -layers completely contained in S, because r ≡ 3 (mod 4). We consider them and one neighboring layer separately in the sum of (3.2) and get The next proposition is stated without a proof because one can use a similar approach as in the proof of Proposition 3.4.

Proposition 3.5.
For any integer numbers r, t with r ≥ 7 and t ≥ 3,

The rooted and corona products of graphs
Given a graph G of order n and a graph H with root vertex v, the rooted product G • v H is defined as the graph obtained from G and H by taking one copy of G and n copies of H, and identifying the ith vertex of G with the vertex v in the ith copy of H for every i ∈ [n]. If H or G is isomorphic to K 1 , then G • v H is equal to G or H, respectively. In this sense, to obtain the rooted product G • v H, hereafter we only consider graphs G and H of order at least two. For every x ∈ V (G), H x will denote the copy of H in G • v H containing x. A formula for the independence number of rooted product graphs can be found in [17].
We need to introduce the following definitions. A near total co-independent dominating set, abbreviated near-TC-ID set, of a graph G, relative to a vertex v, is a set D ⊆ V (G) satisfying the following: (i) v ∈ D; (ii) V (G) − D is an independent set; (iii) every vertex u ∈ D − {v} is adjacent to at least one vertex in D.
The minimum cardinality among all near-TC-ID sets of G relative to v is called the near total co-independent domination number of G relative to v, which we denote as γ nt,coi (G; v). A near-TC-ID set of G relative to v with cardinality γ nt,coi (G; v) is called a γ nt,coi (G; v)-set. Notice that every TC-ID set of G that contains v is a near-TC-ID set of G relative to v.
Next, we present a useful result.
Theorem 4.1. Let G be a graph of order at least three without isolated vertices.
A particular case of the total co-independent domination number of rooted product graphs G • v H, specifically when H contains a universal vertex, is presented below. Let G and H be two graphs of order n G and n H , respectively. The corona product graph G H is defined as the graph obtained from G and H by taking one copy of G and n G copies of H and joining by an edge every vertex from the ith copy of H with the ith vertex of G. Notice that any corona graph G H can be presented as a rooted product graph G • v H , where H ∼ = K 1 ∨ H and v is the vertex of K 1 . Also, observe that α(H ) = α(H). Hence, Lemma 1.3 and Theorem 4.4 lead to the following result.