The Sierpi\'{n}ski Domination Number

Let $G$ and $H$ be graphs and let $f \colon V(G)\rightarrow V(H)$ be a function. The Sierpi\'{n}ski product of $G$ and $H$ with respect to $f$, denoted by $G \otimes _f H$, is defined as the graph on the vertex set $V(G)\times V(H)$, consisting of $|V(G)|$ copies of $H$; for every edge $gg'$ of $G$ there is an edge between copies $gH$ and $g'H$ of $H$ associated with the vertices $g$ and $g'$ of $G$, respectively, of the form $(g,f(g'))(g',f(g))$. In this paper, we define the Sierpi\'{n}ski domination number as the minimum of $\gamma(G\otimes _f H)$ over all functions $f \colon V(G)\rightarrow V(H)$. The upper Sierpi\'{n}ski domination number is defined analogously as the corresponding maximum. After establishing general upper and lower bounds, we determine the upper Sierpi\'{n}ski domination number of the Sierpi\'{n}ski product of two cycles, and determine the lower Sierpi\'{n}ski domination number of the Sierpi\'{n}ski product of two cycles in half of the cases and in the other half cases restrict it to two values.


Introduction
Sierpiński graphs represent a very interesting and widely studied family of graphs.They were introduced in 1997 in the paper [15], where the primary motivation for their introduction was the intrinsic link to the Tower of Hanoi problem, for the latter problem see the book [12].Intensive research of Sierpiński graphs led to a review article [11] in which state of the art up to 2017 is summarized and unified approach to Sierpiński-type graph families is also proposed.Later research on Sierpiński graphs includes [2,3,6,19,23].
Sierpiński graphs have a fractal structure, the basic graphs of which are complete graphs.In 2011, Gravier, Kovše, and Parreau [7] introduced a generalization in such a way that any graph can act as a fundamental graph, and called the resulting graphs generalized Sierpiński graphs.We refer to the papers [1,4,5,13,14,16,17,20,21,22,24] for investigations of generalized Sierpiński graphs in the last few years.
An interesting generalization of Sierpiński graphs in the other direction has recently been proposed by Kovič, Pisanski, Zemljič, and Žitnik in [18].Namely, in the spirit of classical graph products, where the vertex set of a product graph is the Cartesian product of the vertex sets of the factors, they introduced the Sierpiński product of graphs as follows.Let G and H be graphs and let f : V (G) → V (H) be an arbitrary function.The Sierpiński product of graphs G and H with respect to f , denoted by G ⊗ f H, is defined as the graph on the vertex set V (G) × V (H) with edges of two types: • type-1 edge: (g, h)(g, h ′ ) is an edge of G ⊗ f H for every vertex g ∈ V (G) and every edge hh ′ ∈ E(H), • type-2 edge: (g, f (g ′ ))(g ′ , f (g)) is an edge of G ⊗ f H for every edge gg ′ ∈ E(G).
We observe that the edges of type-1 induce n(G) = |V (G)| copies of the graph H in the Sierpiński product G ⊗ f H.For each vertex g ∈ V (G), we let gH be the copy of H corresponding to the vertex g.A type-2 edge joins vertices from different copies of H in G ⊗ f H, and is called a connecting edges of G ⊗ f H.A vertex incident with a connecting edge is called a connecting vertex.We observe that two different copies of H in G ⊗ f H are joined by at most one edge.A copy of the graph H corresponding to a vertex of the graph G in the Sierpiński product G ⊗ f H is called an H-layer.
Let G and H be graphs and H G be the family of functions from V (G) to V (H).We introduce new types of domination, the Sierpiński domination number, denoted by γ S (G, H), as the minimum over all functions f from H G of the domination number of the Sierpiński product with respect to f , and upper Sierpiński domination number, denoted by Γ S (G, H), as the maximum over all functions f ∈ H G of domination number of the Sierpiński product with respect to f .That is, In this paper, we initiate the study of Sierpiński domination in graphs.In Section 1.1 we present the graph theory notation and terminology we follow.In Section 2 we discuss general lower and upper bounds on the (upper) Sierpiński domination number.Our main contribution in this introductory paper is to determine the upper Sierpiński domination number of the Sierpiński product of two cycles, and to determine the lower Sierpiński domination number of the Sierpiński product of two cycles in half of the cases and in the other half cases restrict it to two values.

Notation and terminology
We generally follow the graph theory notation and terminology in the books [8,9,10]  .We denote the path, cycle and complete graph on n vertices by P n , C n , and K n , respectively.For k ≥ 1 an integer, we use the notation [k] = {1, . . ., k} and [k] 0 = {0, 1, . . ., k}.We generally label vertices of the considered graphs by elements of [n].In this case, the mod function over the set [n] is to be understood in a natural way, more formally, we apply the following operation for t ≥ 1: A vertex dominates itself and its neighbors, where two vertices are neighbors in a graph if they are adjacent.A dominating set of a graph G is a set S of vertices of G such that every vertex in G is dominated by a vertex in S. The domination number, γ(G), of G is the minimum cardinality of a dominating set of G.A dominating set of cardinality γ(G) is called a γ-set of G.A thorough treatise on dominating sets can be found in [8,9].If S is a set of vertices in a graph G, then we will use the notation G|S to denote that the vertices in the set S are assumed to be dominated and hence γ(G|S) is the minimum number of vertices in the graph G needed to dominate V (G) \ S. We note that it could be that a vertex in S is still a member of a such a minimum dominating set no matter that we do not need to dominate the vertices in S themselves.If S = {x}, then we simply denote G|S by G|x rather than G|{x}.

General lower and upper bounds
We present in this section general lower and upper bounds on the (upper) Sierpiński domination number.
Proof.Let G ⊗ f H be an arbitrary Sierpiński product of graphs G and H and let X be a γ-set of G ⊗ f H. Assuming for a moment that all the connecting edges are removed from G ⊗ f H, we obtain n(G) disjoint copies of H for which we clearly need n(G)γ(H) vertices in a minimum dominating set.Consider now an arbitrary connecting edge e = (g, f Hence the only situation in which e has an effect on γ(G ⊗ f H) is when (g, f (g ′ )) ∈ X and (g ′ , f (g)) / ∈ X (or the other way around).But in this case, the effect of the presence of the edge e is that because (g, f (g ′ )) dominates one vertex of g ′ H, the edge e might reduce the domination number by 1.That is, each connecting edge can drop the domination number of G ⊗ f H by at most 1, which proves the left inequality.The other two inequalities are clear.
To show that the lower bound of Theorem 2.1 is achieved, we show later in Theorem 3.8 that for n ≥ 3 and k ≥ 1, if we take G = C n and H = C 3k+1 where n ≡ 0 (mod 4), then γ S (G, H) = kn = n(G)γ(H) − m(G).The upper bound of Theorem 2.1 is obtained, for example, for the Sierpiński product of two complete graphs.More generally, to achieve equality in the upper bound of Theorem 2.1 we require the graph H to have the following property.
Theorem 2.2.The equality in Γ S (G, H) ≤ n(G)γ(H) is achieved if and only if there exists a vertex x ∈ V (H) such that γ(H|x) = γ(H).
Proof.Suppose that H has a vertex x that satisfies γ(H|x) = γ(H).In this case, we consider the Sierpiński product G ⊗ f H with the function f : V (G) → V (H) defined by f (v) = x for every vertex v ∈ V (G).Consequently each connecting edge in the product is of the form (g, x)(g ′ , x).Thus, if X is a γ-set of G ⊗ f H, then |X ∩ V (gH)| = γ(H) because the only vertex of gH that can be dominated from outside gH is (g, x), but we have assume that γ(H|x) = γ(H).Therefore, Γ S (G, H) = n(G)γ(H).
For the other implication suppose that γ(H|x) < γ(H) for every vertex x ∈ V (H).For an arbitrary edge To conclude this section we describe large classes of graphs for which the second and the third inequality of Theorem 2.1 are both equality.
Proof.Let G and H be graphs such that ∆(G) < n(H) and γ(H) = 1.Thus, by Theorem 2.1, γ S (G, H) ≤ Γ S (G, H) ≤ n(G) and it is straightforward that the Sierpiński product of graphs G and H can be dominated by taking one dominating vertex from each H-layer to the dominating set.It remains to show that the inequality γ S (G, H) ≥ n(G) also holds.Suppose that γ S (G, H) ≤ n(G) − 1.Let D be a dominating set of G ⊗ f H, where f is such that that minimizes the domination number.Therefore there is an H-layer, denote it by Since there are at most ∆(G) connecting edges incident with vertices from each H-layer and ∆(G) < n(H),then all the vertices from an H-layer cannot be dominated by the vertices from the neighboring layers.Therefore we have and the result follows.

The Sierpiński domination number of cycles
Let us recall firstly the domination number of a path and a cycle.
In this section, we shall prove the following results.
In order to prove Theorems 3.2 and 3.3, we consider three cases, depending on the value of p.

The cycle C n and cycles C 3k+1
To determine Γ S (C n , C 3k+1 ), we prove a slightly more general result.For this purpose, we define a class of graphs H k as follows.
Definition 3.4.For k ≥ 1, let H k be the class of all graphs H that have the following properties. (a) , then there exists a γ-set of H that contains x and y, where x = y is allowed.
We show, for example, that for every k ≥ 1, the cycle C 3k+1 belongs to the class H k .
Proposition 3.5.For k ≥ 1, the class H k of graphs contains the cycle C 3k+1 . Proof.
, property (a) in Definition 3.4 holds.To prove that property (b) in Definition 3.4 holds, let x, y ∈ V (H).Since H is vertex-transitive, every specified vertex belongs to some γ-set of H.In particular, if x = y, then property (b) is immediate.Hence, we may assume that x ̸ = y.Let H be the cycle , where renaming vertices if necessary, we may assume that x = v 1 .Let y = v i , and so i ∈ that is, H ′ is obtained from H by removing x and y, and removing all neighbors of x and y.If H ′ is connected, then H ′ is a path P 3(k−2)+j for some j where j ∈ [3].In this case, γ(H ′ ) = k − 1.If H ′ is disconnected, then H ′ is the disjoint union of two paths P k 1 and P k 2 , where k 1 + k 2 = 3(k − 2) + 1.Thus renaming k 1 and k 2 if necessary, we may assume that either k 1 = 3j 1 and k 2 = 3j 2 + 1 where j 1 ≥ 1, j 2 ≥ 0, and For n ≥ 3 an integer, a circulant graph C n ⟨L⟩ with a given list L ⊆ {1, . . ., ⌊ 1  2 n⌋} is a graph on n vertices in which the ith vertex is adjacent to the (i + j)th and (i − j)th vertices for each j in the list L and where addition is taken modulo n.For example, for n = 3k + 1 where k ≥ 1 and L = {1}, the circulant graph C n ⟨L⟩ is the cycle C 3k+1 , which, by Proposition 3.
By property (a) in Definition 3.4, for every vertex g ∈ V (G), there exists a γ-set of gH that contains the vertex (g, f (g)) = (g, h).If g ∈ D G , let D g be a γ-set of gH that contains the vertex (g, f (g)) = (g, h), and so , and so in this case The set D is a dominating set of G ⊗ f H, and so For the fixed vertex h chosen earlier, we note that the set of vertices (g, h) for all g ∈ V (G) induces a subgraph of G⊗ f H that is isomorphic to G ∼ = C n .We denote this copy of G by Gh.Among all γ-sets of G ⊗ f H, let D * be chosen to contain as many vertices of Gh as possible.
By inequalities ( 1) and ( 2), we have By equation (3), we have Γ Proof.Let f : V (G) → V (H) be an arbitrary function.Let H i be the ith copy of H corresponding to the vertex g i of G for all i ∈ [n].Let D be the dominating set of G ⊗ f H constructed as follows.Let x i y i+1 be the connecting edge from where addition is taken modulo n.Thus, the vertex and y i+1 = (g i+1 , f (g i )).We note that possibly x i = y i .By property (b) in Definition 3.4, there exists a γ-set of H i that contains both x i and y i .For i ∈ [n], we define the sets D i,1 , D i,2 , and D i,3 as follows.Let D i,1 be a γ-set of we define the set D i as follows.
For example, the set D 1 dominates all vertices of H 1 − x 1 .The set D 2 contains the vertex y 2 , which is adjacent to the vertex x 1 of H 1 , and contains the vertex x 2 , which is adjacent to the vertex y 3 of H 3 , implying that D 2 dominates the vertex x 1 of H 1 , all vertices of H 2 , and the vertex y 3 of H 3 .The set D 3 dominates all vertices of H 3 − y 3 .Thus, D 1 ∪ D 2 ∪ D 3 dominates all vertices in V (H 1 )∪V (H 2 )∪V (H 3 ) in the Sierpiński product G⊗ f H.Moreover, If n ≡ 1 (mod 3), then the set D n is a γ-set of H n , and in this case This completes the proof of Claim 2. (2) The proof of Theorem 3.6 follows as an immediate consequence of Claims 1 and 2.
As a consequence of Proposition 3.5, we have the following special case of Theorem 3.6.Proof.For n ≥ 3 and k ≥ 1, let G = C n and let H = C 3k+1 .Let G be the cycle given by g 1 g 2 . . .g n g 1 .We adopt our notation employed in our earlier proofs.For notational convenience, we let V (H) = {1, 2, . . ., 3k + 1} where vertices i and i + 1 are consecutive on the cycle H for all i ∈ [3k + 1] (and where addition is taken modulo 3k + 1, and so vertex 1 and vertex 3k + 1 are adjacent).
As before, we denote the copy g i H of H corresponding to the vertex g i simply by H i for each i ∈ [n].Thus, H i = C 3k+1 is the cycle (g i , 1), (g i , 2), . . ., (g i , 3k + 1), (g i , 1) for all i ∈ [n].Recall that we denote the connecting edge from H i to H i+1 by x i y i+1 for all i ∈ [n], where ), and addition is taken modulo n.Thus, y i = (g i , f (g i−1 )) and By Proposition 3.5, the graph H belongs to the class H k .Thus, γ(H) = k + 1 and γ(H − v) = k for every vertex v ∈ V (H).Furthermore, if x, y ∈ V (H) where x = y is allowed, then there exists a γ-set of H that contains x and y.
Hence for all i ∈ {2, 3, . . ., ℓ − 1}, the vertices x i and y i are at distance 2 in H i , implying that x i ∈ D i .We consider four cases to determine which properties hold for the boundary conditions (that is for i ∈ {1, ℓ}) and finally to set the upper bound on the domination number in each case.
We proceed further with two claims.The first claim establishes a lower bound on γ S (G, H), and the second claim an upper bound on γ S (G, H).Combining these two bounds yields the desired result in the statement of the theorem.
Proof.Let f : V (G) → V (H) be an arbitrary function.We show that Let D be a γ-set of G ⊗ f H constructed, and let If the vertex x i is not dominated by D i , then either x i ̸ = y i , in which case x i is dominated by the vertex y i+1 ∈ D, or x i = y i , in which case x i is dominated by the vertex x i−1 ∈ D or the vertex y i+1 ∈ D. Analogously, if the vertex y i is not dominated by D i , then either x i ̸ = y i , in which case y i is dominated by the vertex x i−1 ∈ D, or x i = y i , in which case y i is dominated by the vertex x i−1 ∈ D or the vertex y i+1 ∈ D. If a vertex is not dominated by D i , then such a vertex is x i or y i , and we say that such a vertex is dominated from outside H i .
Similarly as before, we proceed with a claim that delivers properties of sets D i leading to the desired lower bound on the Sierpiński domination number.
Claim 3.1.The following properties hold in the graph Proof.Suppose that D i contains a vertex w i that dominates x i .Possibly, w i = x i .In order to dominate the 3(k − 1) + 2 vertices in H i not dominated by w i , at least k additional vertices are needed even if the vertex y i is dominated outside the cycle H i .Thus in this case, Suppose that neither x i nor y i is dominated by D i , implying that both x i and y i are dominated from outside the cycle H i .Thus, D i is a dominating set of , and by the minimality of D we have |D i | = γ(P 3k+1 ) = k + 1.Hence, we may assume that x i ̸ = y i .If x i and y i are adjacent, then H ′ i = P 3k , and by the minimality of D we have |D i | = γ(P 3k ) = k.Suppose that x i and y i are not adjacent, and so H ′ is the disjoint union of two paths P k 1 and P k 2 , where k 1 + k 2 = 3k.If k 1 = 3j 1 + 1 and k 2 = 3j 2 + 2 (or if k 1 = 3j 1 + 2 and k 2 = 3j 2 + 1) for some integers j i and j 2 where where addition is taken modulo n.Furthermore in this case when |D i | = k, the vertices x i and y i are distinct and are both dominated from outside H i , implying that y i+1 ∈ D i+1 and x i−1 ∈ D i−1 .This implies that if n is even, then |D| ≥ kn + n/2, and if n is odd, then |D| ≥ kn + (n + 1)/2.This proves inequality (4).Proof.Let f : V (G) → V (H) be the function defined by for all i ∈ [n] where addition is taken modulo n.Adopting our earlier notation, recall that y i = (g i , f (g i−1 )) and Case 1. n ≡ 0 (mod 4).

The cycle C n and cycles C 3k
In this section, we determine the Sierpiński domination number γ S (C n , C 3k ) and the upper Sierpiński domination number Γ S (C n , C 3k ).Theorem 3.11.For n ≥ 3 and k ≥ 1, Proof.We adopt our notation from the earlier sections.Let G ∼ = C n be the cycle g 1 g 2 . . .g n g 1 , and let H i be the ith copy of C 3k corresponding to the vertices g i of G for i ∈ [n].As before, we denote the connecting edge from H i to H i+1 by x i y i+1 for all i ∈ [n].With our assumption that neither x i nor y i is dominated by D i , the set D i is a γ-set of H ′ i = H i − {x i , y i }.If x i = y i , then H ′ i = P 3k−1 , and by the minimality of D we have |D i | = γ(P 3k−1 ) = k.Hence, we may assume that x i ̸ = y i .If x i and y i are adjacent, then H ′ i = P 3k−2 , and by the minimality of D we have |D i | = γ(P 3k−2 ) = k.Suppose that x i and y i are not adjacent, and so H ′ is the disjoint union of two paths P k 1 and P k 2 , where k 1 + k 2 = 3k − 2. If k 1 = 3j 1 + 1 and k 2 = 3j 2 for some integers j i and j 2 where j 1 + j 2 = k − 1, then |D i | = ⌈k 1 /3⌉ + ⌈k 2 /3⌉ = (j 1 + 1) + j 2 = k.Analogously, if k 1 = 3j 1 and k 2 = 3j 2 + 1, then |D i | = k.If k 1 = 3j 1 +2 and k 2 = 3j 2 +2 for some integers j i and j 2 where j 1 +j 2 = k−2, then |D i | = ⌈k 1 /3⌉ + ⌈k 2 /3⌉ = (j 1 + 1) + (j 2 + 1) = k.In all cases, |D i | = k, implying that Since f : V (G) → V (H) was chosen as an arbitrary function, and D as an arbitrary γ-set of G ⊗ f H, we deduce that γ S (C n , C 3k ) = Γ S (C n , C 3k ) = γ(G ⊗ f H) = kn.

Concluding remarks
It seems to us that in the vast majority of cases where the lower Sierpiński domination number of the Sierpiński product of two cycles is specified to two values exactly, the larger of the two is the correct value.However, the following example, which surprised us, demonstrates that there are also cases where the exact value is the smaller of the two possible values.
Let G ∼ = C 18 with V (G) = [18] and let H ∼ = C 7 with V (H) = [7] and let the function f : V (G) → V (H) be defined as follows: Then Theorem 3.2 asserts that γ(G ⊗ f H) ∈ {36, 37} and it is straightforward to check that the exact value is γ(G ⊗ f H) = 36.
on domination in graphs.Specifically, let G be a graph with vertex set V (G) and edge set E(G), and of order n(G) = |V (G)| and size m(G) = |E(G)|.For a subset S of vertices of a graph G, we denote by G − S the graph obtained from G by deleting the vertices in S and all edges incident with vertices in S. If S = {v}, then we simply write G − v rather than G − {v}.The subgraph induced by the set S is denoted by G[S]

Theorem 2 . 1 .
If G and H are graphs, then of H that contains both x and y.Hence, property (b) holds.
the restriction of D * to the copy of G.If a vertex (g, h) / ∈ D * G and (g, h) is not dominated by D * G , then D * g is a γ-set of gH by the minimality of the set D * .However in this case, we could replace the set D * g be a γ-set of gH that contains the vertex (g, h) to produce a new γ-set of G ⊗ f H that contains more vertices from the copy of G than does D * , a contradiction.Hence, the set D * G is a dominating set in the copy of G, and so |D * g (G)| ≥ γ(G).By the minimality of the set D * and by property (a) in Definition 3.4, for each vertex g ∈ V (G), we have |D
y i is dominated by D i , then either d(x i , y i ) ≡ 1 (mod 3), in which case |D i | = k, or d(x i , y i ) ̸ ≡ 1 (mod 3), in which case |D i | = k + 1.This proves properties (a) and (b) of the claim.(2)

Claim 4 . 2
γ S (C n , H) ≤ kn + n Let f : V (G) → V (H) be an arbitrary function.Let D be a γ-set of G ⊗ f H, and letD i = D ∩ V (H i ) for i ∈ [n].We show that |D i | = k for all i ∈ [n].If both vertices x i and y i are dominated by D i , then D i is a γ-set of H i ∼ = C 3k ,and so |D i | = k.If exactly one of x i and y i is dominated by D i , say x i , then by the minimality of the set D, the set D i is a γ-set of H i − y i ∼ = P 3k−1 , and so |D i | = k.Hence, we may assume that neither x i nor y i is dominated by D i , for otherwise, |D i | = k and the desired bound follows.
5, belongs to the class H k .More generally, for n = k(2p+1)+1 where k ≥ 1, p ≥ 1, and L = [p], the circulant graph C n ⟨L⟩ belongs to the class H k .We omit the relatively straightforward proof.These examples of circulant graphs serve to illustrate that for each k ≥ 1, one can construct infinitely many graphs in the class H k .We determine next the upper Sierpiński domination number Γ S (C n , H) of a cycle C n and a graph H in the family H k .
Theorem 3.6.For n ≥ 3 and k ≥ 1, if H ∈ H k , then Γ S (C n , H) = kn + n 3 .Proof.For n ≥ 3 and k ≥ 1, let G ∼ = C n and let H ∈ H k .Let G be the cycle given by g 1 g 2 . . .g n g 1 .In what follows, we adopt the following notation.For each i ∈ [n], we denote the copy g i H of H corresponding to the vertex g i simply by H i .We proceed further with two claims.The first claim establishes a lower bound on Γ S (C n , H), and the second claim establishes an upper bound on Γ S (C n , H).