Some Families of Graphs with Small Power Domination Number

Najibeh Shahbaznejad∗1, Adel P. Kazemi†1, and Ignacio M. Pelayo‡2

1Department of Mathematics, University of Mohaghegh Ardabili, Iran 2Departament de Matem`atiques, Universitat Polit`ecnicade Catalunya, Spain

June 28, 2021

Abstract Let G be a graph with the vertex set V (G) and S be a subset of V (G). Let cl(S) be the set of vertices built from S, by iteratively applying the following propagation rule: if a vertex and all of its neighbors except one of them are in cl(S), then the exceptional neighbor is also in cl(S). A set S is called a zero forcing set of G if cl(S) = V (G). The zero forcing number Z(G) of G is the minimum cardinality of a zero forcing set. Let cl(N[S]) be the set of vertices built from the closed neighborhood N[S] of S, by iteratively applying the previous propagation rule. A set S is called a power dominating set of G if cl(N[S]) = V (G). The power domination number γp(G) of G is the minimum cardinality of a power dominating set. In this paper, we present some families of graphs that their power domination number is 1 or 2. Keywords: domination, power domination, maximum degree, mycieleskian graphs, central graphs, middle graphs. AMS subject classiﬁcation: 05C38, 05C76, 05C90.

1 Introduction

This paper is devoted to the study of both the power domination number of connected graphs introduced in [17]. arXiv:2106.13496v1 [math.CO] 25 Jun 2021 The notion of power domination in graphs is a dynamic version of domination where a set of vertices (power) dominates larger and larger portions of a graph and eventually dominates the whole graph. The introduction of this parameter was mainly inspired by a problem in the electric power system industry [3]. Electric power networks must be continuously monitored. One usual and eﬃcient way of accomplish this monitoring, consist in placing phase measurement units (PMUs), called PMUs, at selected network locations. Due to the high cost of the PMUs, their number must be minimized, while maintaining the ability to monitor (i.e. to observe) the entire network. The power domination problem consists

∗[email protected] †[email protected] ‡[email protected]

1 thus of finding the minimum number of PMUs needed to monitor a given electric power system. In other words, a power dominating set of a graph is a set of vertices that observes every vertex in the graph, following the set of rules for power system monitoring described in [17]. Since it was formally introduced in [17], the power domination number has generated con- siderable interes; see, for example, [5,9, 10, 12, 16, 29]. The defnition of the power domination number leads naturally to the introduction and study of the zero forcing number. As a matter of fact, the zero forcing number of a connected graph G was introduced in [2] as a tight upper bound for the maximum nullity of the set of all real symmetric matrices whose pattern of off-diagonal entries coincides with off-diagonal entries of the adjacency matrix of G, and independently by mathematical physicists studying control of quantum systems [6]. Since then, this parameter has been extensively investigated; see, for example, [8, 11, 14, 15, 18, 20]. In this paper, we present a variety of graph families such that all theirs members have power dominating sets of cardinality at most 2.

1.1 Basic terminology All the graphs considered are undirected, simple, finite and (unless otherwise stated) connected. Let G = (V (G),E(G)) be a graph which V (G) and E(G) are the vertex set and the edge set of G, respectively. Let v be a vertex of a graph G. The open neighborhood of v is NG(v) = {w ∈ V (G): vw ∈ E}, and the closed neighborhood of v is NG[v] = NG(v) ∪ {v} (we will write N(v) and N[v] if the graph G is clear from the context). The degree of v is deg(v) = |N(v)|. The minimum degree (resp. maximum degree) of G is δ(G) = min{deg(u): u ∈ V (G)} (resp. ∆(G) = max{deg(u): u ∈ V (G)}). If deg(v) = 1, then v is said to be a leaf of G. The distance between vertices v, w ∈ V (G) is denoted by dG(v, w), or d(v, w) if the graph G is clear from the context. The diameter of G is diam(G) = max{d(v, w): v, w ∈ V (G)}. Let W ⊆ V (G). The open neighborhood of W is N(W ) = ∪v∈W N(v) and the closed neighborhood of W is N[W ] = ∪v∈W N[v]. Let u, v ∈ V (G) be a pair of vertices such that d(u, w) = d(v, w) for all w ∈ V (G) \{u, v}, i.e., either N(u) = N(v) or N[u] = N[v]. In both cases, u and v are said to be twins. Let H and G be a pair of graphs. The graph H is called a subgraph of G if it can be obtained from G by removing some of the edges and vertices. Moreover, a subgraph H of G is called induced if it can be obtained from G by removing some vertices. An induced subgraph of G made by W , a subset of V (G), is denoted by G[W ], and its edge set is {vw ∈ E(G): v ∈ W, w ∈ W }. The graph H is a minor of G if it can be obtained from G by removing some vertices and by removing and contracting edges. Let D be a subset of V (G). D is called a dominating set whenever N[D] = V (G). Also, the domination number γ(G) is the cardinality of smallest dominating set for G. Let Kn, Kh,n−h, K1,n−1, Pn, Wn and Cn denote complete graph, complete bipartite graph, star, path, wheel and cycle, respectively, which the order of each is n. For undefined terminology and notation, we refer the reader to [7]. The remainder of this paper is organized into five more sections as follows. Continuing this section is devoted to introducing the zero forcing sets, the zero forzing number Z(G) of a connected graph G, power dominating sets and the power domination number γp(G) of a connected graph G are first introduced. In Section 2, we present a brief description of known and new results. In Section 3, some contributions involving graphs with high maximum degree are presented. Finally, in Section 4 to 6, some results are presented for mycieleskian graphs, central graphs and middle graphs. The concept of zero forcing can be described via the following coloring game on the vertices of a given graph G = (V,E). Let U be a proper subset of V . The elements of U are colored black, meanwhile the vertices of W = V \ U are colored white. The color change rule is:

If u ∈ U and exactly one neighbor w of u is white, then change the color of w to black.

2 In such a case, we denote this by u → w, and we say, equivalentely, that u forces w, that u is a forcing vertex of w and also that u → w is a force. The closure of U, denoted cl(U), is the set of black vertices obtained after the color change rule is applied until no new vertex can be forced; it can be shown that cl(U) is uniquely determined by U (see [2]).

Deﬁnition 1 ([2]). A set U ⊆ V (G) is called a zero forcing set of G if cl(U) = V (G).

A minimum zero forcing set, a ZF-set for short, is a zero forcing set of minimum cardinality. The zero forcing number of G , denoted by Z(G), is the cardinality of a ZF-set. A chronological list of forces FU associated with a set U is a sequence of forces applied to obtain cl(U) in the order they are applied. A forcing chain for the chronological list of forces FU is a maximal sequence of vertices (v1, ..., vk) such that the force vi → vi+1 is in FU for 1 ≤ i ≤ k − 1. Each forcing chain induces a distinct path in G, one of whose endpoints is in U; the other is called a terminal. Notice that a zero forcing chain can consist of a single vertex (v1), and this happens if v1 ∈ U and v1 does not perform a force. Observe also that any two forcing chains are disjoint.

Proposition 2 ([11]). Let G be a graph of order n. Then, Z(G) = 1 if and only if G is the path Pn.

A graph is outerplanar if it has a crossing-free embedding in the plane such that all vertices are on the same face. The path cover number P (G) of a graph G is the smallest positive integer k such that there are k [ k vertex-disjoint induced paths P1,...,Pk in G that cover all the vertices of G, i.e., V (G) = V (Pi). i=1 Proposition 3 ([4]). For any graph G, P (G) ≤ Z(G).

Theorem 4 ([21]). Let G be a graph of order n ≥ 5. Then, Z(G) = 2 if and only if G is an outerplanar graph with P (G) = 2.

Zero forcing is closely related to power domination, because power domination can be described as a domination step followed by the zero forcing process or, equivalentely, zero forcing can be described as power domination without the domination step. In other words, the power domination process on a graph G can be described as choosing a set S ⊂ V (G) and applying the zero forcing process to the closed neighbourhood N[S] of S. The set S is thus a power dominating set of G if and only if N[S] is a zero forcing set for G.

Deﬁnition 5 ([17]). Let G be a graph and S ⊆ V (G). The set S is called a power dominating set of G if cl(N[S]) = V (G).

A minimum power dominating set, a PD-set for short, is a power dominating set of minimum cardinality. The power dominating number of G , denoted by γp(G), is the cardinality of a PD-set.

Deﬁnition 6 ([26]). If G is a graph and S0 = S ⊆ V (G), then the sets Si(i > 0) of vertices monitored by S0 at step i are as follows: S1 = NG[S0] (domination step), and S Si+1 = {NG[v]: v ∈ Si such that|NG[v] \ Si| ≤ 1 (propagation steps).

2 Basic Results

As a straight consequence of these deﬁnitions, it is derived both that γp(G) ≤ Z(G) and γp(G) ≤ γ(G). Moreover, this pair of inequalities along with Theorem4, allow us to derive the following results.

Corollary 7. Let G be a graph of order n.

• If G is outerplanar and P (G) = 2, then γp(G) ≤ 2.

3 • ∆(G) = n − 1 if and only if γp(G) = γ(G) = 1.

We end this section by presenting a ﬁrst list of new and known results involving this parameter along with a Table containing information of some basic graph families.

Proposition 8. If G is a connected graph of order al most 5, then γp(G) = 1. Moreover,

• The smallest connected graph G such that γp(G) = 2 is the H-graph (see Figure1 (a)).

• Three of the smallest connected graph G with no twin vertices such that γp(G) = 2 are in the Figure 1 (b), (c) and (d).

(a) (b) (c) (d)

Figure 1: Some small graphs

G Pn Cn Kn K1,n K2,n Kh,n−h Wn γp(G) 1 1 1 1 1 2 1 n+2 n+2 γ(G) b 3 c b 3 c 1 1 2 2 1 Z(G) 1 2 n − 1 n − 2 n − 2 n − 2 3

Table 1: Power domination, domination and zero forcing numbers of some basic graph families.

Proposition 9. Let G = Kr1,··· ,rk be the complete k-partite graph with 2 ≤ k and 1 ≤ r1 ≤ r2 ≤ · · · ≤ rk k and V (G) = ∪i=1Vi. Let Ge the graph obtained from G by deleting an edge e = vw ∈ E(G). Then

(1) If r1 ≤ 2, then γp(G) = 1.

(2) If r1 ≥ 3, then γp(G) = 2.

(3) If r1 ≤ 2, then γp(Ge) = 1. 1, if {v, w} ∩ V =6 ∅ (4) If r = 3, then γ (G ) = 1 1 p e 2, otherwise.

(5) If 4 ≤ r1, then γp(Ge) = 2.

Proof. (1) Take v1 ∈ V1. Notice that N[v1] = V (G) \ [V1 − v1]. If r1 = 1, then {v1} is a dominating set 0 0 of G, i.e., γp(G) = 1. Suppose that r1 = 2 and V1 = {v1, v1}. Then, for any vertex u 6∈ V1, u → v1, 0 which means that γp(G) = 1, as N[v1] = V (G) \{v1}.

(2) For every u ∈ Vi, N[u] = V (G) \ [Vi − u]. Thus, γp(G) ≥ 2. Take S = {v1, v2}, where v1 ∈ V1 and v2 ∈ V2. Notice that N[S] = V (G). Hence, γp(G) = γ(G) = 2.

4 (3) If {v, w} ∩ V1 = ∅, then proceed as in item (1). Suppose w.l.o.g. that v ∈ V1. Notice that N[v] = V (G) \ [(V1 − v) ∪ {w}]. If r1 = 1, then for every u 6∈ {v, w}, u → w. Thus, γp(Ge) = 1. Otherwise, 0 0 0 suppose that r1 = 2 and V1 = {v, v }. Then, N[v ] = V (G) − v and for any vertex u 6∈ {v, v , w}, u → v. Hence, γp(Ge) = 1.

(4) If {v, w} ∩ V1 = ∅, then proceed as in item (2). Otherwise, suppose w.l.o.g. that v ∈ V1 and 0 00 0 00 00 V1 = {v, v , v }. Notice that N[v ] = V (G) \{v, v }. Next, observe that w → v and for any vertex 0 00 u 6∈ {v, v , v , w}, u → v. Hence, γp(Ge) = 1.

(5) Notice that, for every u ∈ V (G), cl(u) = N[u] and |N[u]| ≤ n − 3. Thus, γp(G) ≥ 2. Moreover, for every pair of vertices {u1, u2} such that u1 ∈ Vi and u2 ∈ Vj and {u1, u2} ∩ {v, w} = ∅, N[{u1, u2}] = V (G). Hence, γp(Ge) = γ(Ge) = 2.

A tree is called a spider if it has a unique vertex of degree greater than 2. We deﬁne the spider number of a tree T , denoted by sp(T ), to be the minimum number of subsets into which V (T ) can be partitioned so that each subset induces a spider.

Theorem 10 ([17]). For any tree T , γp(T ) = sp(T ).

Corollary 11. For any tree T , γp(T ) = 1 if and only if T is a spider. Theorem 12 ([28]). If G is a planar (resp. outerplanar) graph of diameter at most 2 (resp. at most 3), then γp(G) ≤ 2 (resp. γp(G) = 1).

3 Graphs with large maximum degree

Proposition 13. Let G a graph of order n and maximum degree ∆.

(1) If n − 2 ≤ ∆ ≤ n − 1, then γp(G) = 1.

(2) If n − 4 ≤ ∆ ≤ n − 3, then 1 ≤ γp(G) ≤ 2. Proof. Let u a vertex such that deg(u) = ∆, that is, such that |N[u]| = ∆ + 1.

(1) If ∆ = n − 1, then 1 ≤ γp(G) ≤ γ(G) = 1, which means that γp(G) = 1. Let u a vertex such that deg(u) = ∆, that is, such that |N[u]| = ∆ + 1. If ∆ = n − 2, then |N[u]| = n − 1, i.e., there exists a vertex w such that V (G) \ N[u] = {w}. Thus, for some vertex v ∈ N(u), v → w, which means that {u} is a PD-set.

(2) Suppose that ∆ = n − 3. Let w1, w2 ∈ V (G) such that V (G) \ N[u] = {w1, w2}. Take the set S = {u, w1}. If w1w2 ∈ E(G), then S is a dominating set of G, and thus it is a power dominating set. If w1w2 6∈ E(G), then N[S] = V (G) \{w2}. Hence, S is a power dominating set since for some vertex v ∈ N(u), v → w2. Finally, assume that ∆ = n − 4. Let w1, w2, w3 ∈ V (G) such that N = V (G) \ N[u] = {w1, w2, w3}. We distinguish cases.

Case 1: G[N] is not the empty graph K3. Suppose w.l.o.g. that w1w2 ∈ E(G). Take the set S = {u, w1}. If w1w3 ∈ E(G), then S is a dominating set of G, and thus it is a power dominating set. If w1w3 6∈ E(G), then N[S] = V (G) \{w3}. Hence, S is a power dominating set since either w2 → w3 or, for some vertex v ∈ N(u), v → w3.

Case 2: G[N] is the empty graph K3. For i ∈ {1, 2, 3}, let vi ∈ N(u) be such that viwi ∈ E(G). If for every i ∈ {1, 2, 3}, N(vi) ∩ {w1, w2, w3} = {wi}, then {u} is a dominating set of G, and thus it is a power dominating set. If for some i ∈ {1, 2, 3}, |N(vi) ∩ {w1, w2, w3}| ≥ 2, assume w.l.o.g. that i = 1. In this case, S = {u, v1} is a power dominating set since V (G) \{w3} ⊆ N[S] and either v1 → w3 or v3 → w3.

5 There are graphs with maximum degree ∆ = n − 5 such that γp(G) ≥ 3. The simplest example is shown in Figure2.

Figure 2: A graph with maximum degree ∆ = n − 5 and γp(G) = 3

Lemma 14. Let G be a graph of order n ≥ 4. Let u, w1, w2 ∈ V (G) such that deg(u) = n − 3 and V (G) = N[u] ∪ {w1, w2} Then, {u} is a PD-set if and only if w1 and w2 are not twins.

Proof. Suppose ﬁrst that w1 and w2 are twins. In this case, every power dominating set must contain either w1 or w2. Conversely, assume that w1 and w2 are not twins. If N(w1) = {w2}, then for some vertex v ∈ N(u), v → w1 and w1 → w2, which means that {u} is a PD-set. If deg(w1) ≥ 2, then take a vertex v1 ∈ N(u) such that w1 ∈ N(v1) and w2 6∈ N(v1). Thus, v1 → w1 and v2 → w2, for any vertex v2 such that w2 ∈ N(v2).

Corollary 15. Let G be a graph of order n ≥ 4. If there exists a vertex u ∈ V (G) such that deg(u) = n−3 and the pair of vertices of V (G) \ N[u] are not twins, then γp(G) = 1. The converse of this statement is not true. For example, if we consider the graph G displayed in Figure3, then it is easy to check that {w1} is a PD-set of G.

v v 1 v4 v2 5 v3

w1 w2

Figure 3: The list a forcing chains for FN[w1] is: {(w1), (w2), (v2, u, v6), (v1, v4), (v3, v5)}.

Theorem 16. Let G be a (n − 3)-regular graph of order n ≥ 5. Then, γp(G) = 1 if and only if there exist an edge e = uv ∈ E(G) such that | N[v] \ N[u] |= 1. Otherwise γp(G) = 2. ∼ Proof. According to part 2 of proposition 13 γp(G) ≤ 2. If n = 5, then G = C5, and the equivalence is obvious. Suppose thus that n ≥ 6. (⇒): Let S = {u} be a γp-set of G. Let W = V (G) \ N[u] = {x, y}. As S is a γp-set, there must exist a vertex v ∈ N(u) such that |N(v) ∩ W | = 1. Hence, there exist a unique vertex w ∈ N(u) \{v} such that w 6∈ N(v), as deg(v) = n − 3.

6 (⇐): Take the sets S = {u} and W = V (G) \ N[u] = {x, y}. As | N[v] \ N[u] |= 1 and deg(u) = deg(v) = n − 3, |N(v) ∩ W | = 1. Hence, if for example N(v) ∩ W = {x}, then v → x, which means that S is a γp-set of G.

Here we deﬁne two families of graphs; • Let three vertices u and w and z induce a path in G(w is in the middle vertex) such that u or z is the diﬀerence with w in exactly one neighbor, i.e.,

∃x ∈ {u, z} s.t | N[w] \ N[x] |= 1

This induce subgraph is called P3. • For another family, suppose three vertices u and w and z induce a cycle in G such that at least two of them are the diﬀerence with w in exactly one neighbor, i.e.,

∃x, y ∈ {u, w, z} s.t | N[x] \ N[y] |= 1

This induce subgraph is called C3.

Theorem 17. Let G be a (n − 4)-regular graph of order n ≥ 6. γp(G) = 1 if and only if G contains P3 or C3 as a induce subgraph.

Proof. (⇒):Let S = {v} be a γp-set of G and V (G) \ N[v] = {u, w, z}. Now we study four states;

• There is a path between u, w and z. • only two of {u, w, z} are adjacent. • (u, w), (w, z) and (z, u) ∈ E(G). • None of them is adjacent.

In the First state, Without losing generality, let (u, w) and (w, z) ∈ E(G). w is not adjacent to two neighbors of v, which we call w1 and w2(because G is a (n − 4)−regular graph and every vertex is not adjacent to 3 vertices). u and z also is not adjacent to one neighbor of v called u1 and z1. Now we have ﬁve cases;

Case 1: u1 = z1. So u and z are twin, then should u or z ∈ S. therefore γp(G) > 1 and this is a contradiction. Case 2: z1 = w1 (or u1 = w1 ). So z1 → u and next u → w. Then γp(G) = 1 in this case. Case 3: z1 = w1 and u1 = w2. This case is like the previous case. Case 4: None of them is not equal. In this case, each of neighbors of v is adjacent to at least two of {u, w, z}. So we can not continue. Then γp(G) > 1 and so this case is not accepted. Case 5: z1 = u1 = w1. z1 = u1 = w1 is not adjacent to any one of {u, w, z} and other neighbors of v are adjacent to at least two of {u, w, z}. So we can not continue and then γp(G) > 1. Thus only the second and third cases in the ﬁrst state are acceptable that in both of them, {u, w, z} induces P3 in G. In the second state, Without losing generality, let (u, w) ∈ E(G). So N(z) = N(v) but u and w are not adjacent to one neighbor of v called u1 and w1. If u1 = w1, then u and w are twin and we can not continue. Now if u1 =6 w1, then In this case, each of neighbors of v are adjacent to at least two of {u, w, z}. So we can not continue propagation. So in both cases, γp(G) > 1. In the third state, u, w and z are not adjacent to two neighbors of v, which we call u1, u2, w1, w2, z1 and z2 respectively. Now we have eight cases;

Case 1: None of them is not equal. Case 2: w1 = u1 = z1 and w2 = u2 = z2.

7 Case 3: w1 = u1 = z1 and w2 = u2. Case 4: w1 = u1 = z1. Case 5: w1 = u1. Case 6: w1 = u1 and u2 = z2. Case 7: w1 = u1 and w2 = u2. Case 8: w1 = u1 and w2 = z2 and z1 = u2. Like argument of the second state, it can easily be seen that there are contradiction in the second, third, fourth and seventh cases. Also {u, w, z} induses C3 in the ﬁfth and sixth and eighth cases. In the fourth state, In this case, u, w, z are twin pairwise. Then γp(G) > 1. (⇐):It is clear.

An example for the fourth state is K4, 4, ··· , 4 that γp(K4, 4, ··· , 4) = 2.

| {zn } | {zn } 4 4 The following corollary is obtained directly from the above theorem.

Corollary 18. Let G be a (n − 4)-regular graph of order n ≥ 6. γp(G) = 2 if and only if G contains neither P3 nor C3 as a induce subgraph.

We know that ” in what condition is γp of a graph equal to 1 ” is a open problem. In the next sections, we obtain it for some families of graphs.

4 Mycieleskian graphs

Deﬁnition 19 ([25]). Let G = (V,E) be a graph with the vertex set V = {vi|1 ≤ i ≤ n}. The Mycieleskian graph µ(G) of a graph G is a graph with the vertex set V ∪ U ∪ {w} such that U = {ui|1 ≤ i ≤ n}, and the edge set E ∪ {uivj|vivj ∈ E(G)} ∪ {uiw|ui ∈ U}.

Theorem 20. [25] If G be a connected graph, then γp(µ(G)) ∈ {1, γG, γp(G) + 1}.

Theorem 21. [25] Let G be a connected graph. If G has one universal vertex, then γp(µ(G)) = 1.

Notice that according to theorem 23, inverse of theorem 21 is not true. Next, we calculate power domination number of mycieleskian of the graphs in the table 1. In the ﬁrst, you can see the direct result of the above theorem.

Corollary 22. γp(µ(Kn)) = γp(µ(K1,n)) = γp(µ(Wn)) = 1

Theorem 23. γp(µ(Pn)) = 1.

Proof. Let G = Pn and V (G) = {v1, ··· , vn}. We claim that S = {v2} is a PD-set for M(G) because {v1, v3, u1, u3} are monitored in d.s(domination step) and {w, u2} are monitored in p.s(propagation steps). Also {v4}, {u4}, {v5}, {u5}, ··· , {vn}, {un} are monitored in next propagation steps.

Theorem 24. If G = C , and n ( 1 n = 3 γp(µ(G)) = 2 n ≥ 4.

Proof. Let G = Cn and V (G) = {v1, ··· , vn}. For n = 3, according to theorem 21 γp(µ(G)) = 1. For n ≥ 4, S = {v1, w} is a PD-set for µ(G) so γp(µ(G)) ≤ 2. Now we prove γp(µ(G)) =6 1. If γp(µ(G)) = 1, then we have three state; State 1: S = {vi}. State 2: S = {ui}. State 3: S = {w}.

8 In state 1, {vi−1, vi+1, ui−1, ui+1} are monitored in d.s but p.s can not occur because all of {vi−1, vi+1, ui−1, ui+1} are adjacent to at least two vertices of V (M(G)) \{vi−1, vi+1, ui−1, ui+1}. In state 2, {w, vi−1, vi+1} are monitored in d.s and again like state 1, p.s can not occur. In state 3, {u1, ··· , un} are monitored in d.s but p.s can not occur. So γp(µ(G)) =6 1 and then γp(µ(G)) = 2.

Theorem 25. γp(µ(Kh,n−h)) = 2, for h ≥ 2, n ≥ 4.

Proof. Let G = Kh,n−h and V (G) = {v1, ··· , vh} ∪ {vh+1, ··· , vn}. If vi ∈ {v1, ··· , vh} and vj ∈ {vh+1, ··· , vn}, then S = {vi, vj} is a PD-set for M(G) because V (µ(G)) \{w} are monitored in d.s and w is monitored in p.s. So γp(µ(G)) ≤ 2. Now we claim γp(µ(G)) =6 1. If γp(µ(G)) = 1, then we have ﬁve state; State 1: S = {vi}, for 1 ≤ i ≤ h. State 2: S = {vj}, for h + 1 ≤ i ≤ n. State 3: S = {ui}, for 1 ≤ i ≤ h. State 4: S = {uj}, for h + 1 ≤ i ≤ n. State 5: S = {w}. In all state, p.s can not occur. Therefore γp(µ(G)) = 2.

A family graph closely related to Mycieleskian graph is called the shadow graph. The shadow graph S(G) of a graph G is the graph obtained from G by adding a new vertex ui for each vertex vi of G and joining ui to the neighbors of vi in G [13]. The vertex ui is called the shadow vertex of vi and we have called the vertex vi the orginal vertex of ui.

Theorem 26. If G be a connected graph, then γp(G) ≤ γp(S(G)) ≤ 2γp(G).

Proof. Let G is a graph with V (G) = {v1, ··· , vn} and V (S(G)) = V (G)∪{u1, ··· , un}. If SS(G) = Y ∪W 0 is a γp-set for S(G) such that Y ⊆ V (G) and W ⊆ {u1, ··· , un}, then SG = Y ∪ Y is a PD-set for G 0 0 that Y ⊆ V (G) and Y contains the orginal vertices of W . So γp(G) ≤ γp(S(G)). 0 0 In other hand, if SG be a γp-set for G, then SS(G) = SG ∪ SG is a PD-set for S(G) that SG contains the shadow vertices of SG.

vil vj1 vj2 vj3 vj(k−2) vj(k−1) vjk

uil ujk uj1 uj2 uj3 uj(k−2) uj(k−1)

Figure 4: Part of the graph G

because suppose that vi ∈ SG and N(vi) = {vi1 , ··· , vik }. in d.s, N(vi) ∪ {ui1 , ··· , uik } are mon-

itored in S(G). Now according to ﬁgure4, if propagation accures from vil and forcing chain be as

(vil , vj1 , ··· , vjk ) in G, then propagation accures from uil and forcing chains are as (uil , vj1 ), (vil , uj1 , vj2 ),

(vj1 , uj2 , vj3 ), ··· ,(vj(k−2) , uj(k−1) , vjk ). Thus γp(S(G)) ≤ 2γp(G).

Lemma 27. If S = {vi1 , ··· , vik } be a γp-set for a graph G in which every vertex vi;

(1) has been at least one neighbor vj ∈ {V (G) \ S} such that N[vj] ⊆ N[vi] or,

(2) has been at least one neighbor vj ∈ S, then γp(G) = γp(S(G)).

9 Proof. If S = {vi1 , ··· , vik } is a γp-set for G and vil ∈ S such that it has been one neighbor vj ∈

{V (G) \ S}, then in p.s, uil is monitored by vj(uil is the shadow vertex of vil ) in S(G).

In other hand, if vil has been on neighbor vj ∈ S, then in d.s uil is monitored by vj ∈ S(G). So like the proof of theorem 26, we have γp(S(G)) ≤ γp(G) and then γp(S(G)) = γp(G).

Corollary 28. If S = {v1, ··· , vk} be a γp-set for a graph G and A ⊆ S that every vertex vi in A;

(1) has been any neighbor vj ∈ {V (G) \ S} such that N[vj] ⊆ N[vi] or,

(2) has been any neighbor vj ∈ S, then γp(S(G)) ≤ 2γp(G)− | A |.

Theorem 29. Let G be a connected graph. If G has one universal vertex, then γp(S(G)) = 1.

Proof. Let G is a graph that v1 is an universal vertex of G and also V (G) = {v1, ··· , vn}. We claim S = {v1} is a γp-set for S(G), because V (S(G)) \{u1} are monitored in d.s and {u1} is monitored in p.s.

In the following, you can see the direct result of the above theorem.

Corollary 30. γp(S(Kn)) = γp(S(K1,n)) = γp(S(Wn)) = 1

Theorem 31. γp(S(Pn)) = 1

Proof. Let G = Pn that V (G) = {v1, ··· , vn}. We claim that S = {v2} is a γp-set for S(G) because {v1, v3, u1, u3} are monitored in d.s and other vertices are monitored in several propagation steps.

Theorem 32. If G = C , then n ( 1 n = 3 γp(S(G)) = 2 n ≥ 4.

Proof. Let G = Cn and V (G) = {v1, ··· , vn}. For n = 3, according to theorem 29 γp(S(C3)) = 1. For n ≥ 4, S = {v1, u1} is a PD-set for S(G) so γp(S(G)) ≤ 2. Now we prove γp(S(G)) =6 1. If γp(S(G)) = 1, then we have two state; State 1: S = {vi}, for 1 ≤ i ≤ n. State 2: S = {ui}, for 1 ≤ i ≤ n. In state 1, {vi−1, vi+1, ui−1, ui+1} are monitored in d.s and {vi−2, vi+2} are monitored in p.s and p.s can not occur because all of {vi−2, vi+2, vi−1, vi+1, ui−1, ui+1} are adjacent to at least two vertices of V (S(G)) \{vi−2, vi+2, vi−1, vi+1, ui−1, ui+1}. In state 2, {vi−1, vi+1} are monitored in d.s but p.s can not occur.

Theorem 33. γp(S(Kh,n−h)) = 2 , for h ≥ 2, n ≥ 4.

Proof. Let G = Kh,n−h and V (G) = {v1, ··· , vh} ∪ {vh+1, ··· , vn}. If vi ∈ {v1, ··· , vh} and vj ∈ {vh+1, ··· , vn}, then S = {vi, vj} is a PD-set for S(G) because V (S(G)) are monitored in d.s. So γp(S(G)) ≤ 2. Now we claim γp(S(G)) =6 1. If γp(S(G)) = 1, then we have four state; State 1: S = {vi}, for 1 ≤ i ≤ h. State 2: S = {vj}, for h + 1 ≤ i ≤ n. State 3: S = {ui}, for 1 ≤ i ≤ h. State 4: S = {uj}, for h + 1 ≤ i ≤ n. In all state, it is clear that all vertices are not monitored. Therefore γp(S(G)) = 2.

10 5 Central graphs

Deﬁnition 34 ([27]). The central graph C(G) of a graph G of order n and size m is a graph of order n n + m and size (2 ) + m which is obtained by subdividing each edge of G exactly once and joining all the non-adjacent vertices of G in C(G). The anti-cycle vertex v of a graph G is a vertex such that G[V (G) \{v}] is an acyclic (a graph having no cycle). Note that G[V (G) \{v}] can be connected or disconnected.

Theorem 35. Let G is connected graph. If G has at least one anti-cycle vertex, then γp(C(G)) = 1.

Proof. Let G be a graph that V (G) = {v1, ··· , vn} and

V (C(G)) = {v1, ··· , vn} ∪ {ui,j | vivj ∈ E(G) for some 1 ≤ i, j ≤ n} and also vk be an anti-cycle vertex of G. We prove that S = {vk} is a PD-set for C(G). At the ﬁrst, we consider the state that V (G)\{vk} is connected because when it isn’t connected will also be also proven like this state. Now let degG(vk) = s and without losing generality suppose that NG(vk) = {v1, v2, ··· , vs}, so vk is not adjacent to {vs+1, ··· , vn}. In the other hand

NC(G)[vk] = {v1, v2, ··· , vs, vk, uk(s+1), ··· , ukn}, then {vs+1, ··· , vn} are monitored in p.s. Notice that V (G) \{vk} induce a tree and all of V (G) \{vk} are monitored. So the propagation can easily be continued from the leaves. Thus in the next p.s, {uij}i,j (for some 1 ≤ i, j ≤ n and i, j =6 k) are monitored.

Inverse of theorem 34 is not true. For example K3,3 has no anti-cycle vertex but γp(C(K3,3)) = 1(proved in theorem 37). Next, we calculate power domination number of central of the graphs in the table 1. In the following, you can see the direct result of theorem 35.

Corollary 36. γp(C(Pn)) = γp(C(Cn)) = γp(C(K1,n)) = γp(C(K2,n)) = 1

Theorem 37. γp(C(Kh,n−h)) = 1 , for h ≥ 3, n ≥ 6.

Proof. Let G = Kh,n−h and V (G) = {v1, ··· , vh} ∪ {vh+1, ··· , vn}. S = {vk} is a γp-set for C(G), such that vi is an arbitrary vertex of G, bacause {v1, ··· , vh} and {vh+1, ··· , vn} induce Kh and Kn−h , respectively, in C(G). If vk ∈ {v1, ··· , vk}, then

{v1, ··· , vk} ∪ {ui,j | vivj ∈ E(G) for 1 ≤ i ≤ k and h + 1 ≤ j ≤ n} are monitored in d.s. Finally {vh+1, ··· , vn} are monitored in p.s.

Theorem 38. γp(C(Wn)) = 2.

Proof. Let G = Wn and V (G) = {v1, ··· , vn} such that v1 be the universal vertex in G and also V (C(G)) = V (G)∪{ui,j | vivj ∈ E(G) for some 1 ≤ i, j ≤ n}. In the ﬁrst, we prove that γp(C(G)) =6 1. If γp(C(G)) = 1, then we have three states for γp-set of C(G); State 1: γp(C(G)) = {uij} for some 1 ≤ i, j ≤ n. State 2: γp(C(G)) = {v1}. State 3: γp(C(G)) = {vi} such that i =6 1. In state 1, {vi, vj} are monitored in d.s but p.s can not occur because degC(G)(vi) = degC(G)(vj) = n ≥ 4. In state 2, {u12, u13, ··· , u1n} are monitored in d.s and {v2, ··· , vn} are also monitored in p.s but p.s can not continue, because every vertex of {v2, ··· , vn} adjacent to two vertices of {uij}i,j6=1. In state 3, without losing generality let S = {v2}. So {u21, u23, u2n, v4, v5, ··· , vn−1} are monitored in d.s and {v1, v3, vn} are monitored in p.s. The propagation can not continue because every vertex of {v3, ··· , vn} adjacent to at least two vertices of {uij}i,j6=2. It can easily be seen that {v1, vi}(2 ≤ i ≤ n) is a γp-set for C(G). So γp(C(G)) = 2.

11 Theorem 39. γp(C(Kn)) = n − 2

Proof. Let G = Kn and V (G) = {v1, ··· , vn} and also V (C(G)) = V (G) ∪ {ui,j | vivj ∈ E(G), 1 ≤ i, j ≤ n}. Suppose that S is a γp-set of C(G)), we prove that if | S |= n − 3, then all vertices of C(G)) are not monitored. In the best state, S can be considered as {v1, ··· , vn−3}. In d.s, {uij}i,j(such that i, j∈ / {n, n − 1, n − 2}) are monitored and in p.d, all vertices of C(G) are monitored except {u(n−2)n, u(n−2)(n−1), u(n−1)n}. It is clear that the propagation can not continue, so | S |> n − 3. In the other hand, If S = {v1, ··· , vn−2}, then all vertices of C(G) are monitored in d.s and p.s.

6 Middle graphs

Deﬁnition 40 ([23]). The middle graph M(G) of a graph G whose vertex set is V (G) ∪ E(G) where two vertices are adjacent if and only if they are either adjacent edges of G or one is a vertex and the other is an edge incident with it.

n−1 Theorem 41 ([23]). γp(M(Pn)) = d 3 e. n Theorem 42 ([23]). γp(M(Cn)) = d 3 e.

Theorem 43 ([23]). γp(M(K1,n)) = 1

Deﬁnition 44 ([24]). Let G = (V,E) be a graph. A subset F ⊆ E is an edge dominating set if each edge in E is either in F or is adjacent to an edge in F . An edge dominating set F is called a minimal edge dominating set (or MEDS) if no proper subset F 0 of F is an edge dominating set. The edge domination 0 number γ (G) is the minimum cardinality among all minimal edge dominating sets.

0 Theorem 45. For any graph G, γp(M(G)) ≤ γ (G).

Proof. Let G be a graph and V (G) = {v1, ··· , vn} and E(G) = {uij | vi and vj are adjacent}. Without 0 losing generality, suppose that u12 is in γ -set of G and it dominate {u1i1 , ··· , u1is , u2j1 , ··· , u2jk } in

G. So if u12 be in γp-set of M(G), then it monitore {u1i1 , ··· , u1is , u2j1 , ··· , u2jk } ∪ {v1, v2} in d.s and 0 {vi1 , ··· , vis , vj1 , ··· , vjk } are monitored in p.s. Therefore if all vertices of γ -set be as a γp-set of M(G), 0 then all vertices of M(G) are monitored. So γp(M(G)) ≤ γ (G).

Theorem 46. For any graph G, γp(M(G)) = 1 if and only if G has one universal edge(the edge that is adjacent to all other edges).

Proof. Let G is a graph and V (G) = {v1, ··· , vn} and E(G) = {uij | vi and vj are adjacent}. 0 (⇐) If G has one universal edge i.e. γ = 1, then according to theorem 45, γp(M(G)) = 1. (⇒) Let γp(M(G)) = {v}. We have two state; State 1: v ∈ V (G). State 1: v ∈ E(G).

In state 1, without losing generality, let v = v1 and NG(v1) = {vj1 , ··· , vjl }. So {u1j1 , ··· , u1jl } are monitored in d.s but propagation can occur unless all vertices of NG(v1) have order 1. So in this state, G can only be a K1,n and we know K1,n has the universal edge. In state 2, without losing generality, let v = u12. (proof by contradiction) Suppose that v1v2 is not an universal edge in G. So there is vivj ∈ E(G) such that i, j∈ / {1, 2}. If vivj be a leaf and vj adjacent to v1(or v2) in G, then u1j adjacent to {u12, uij, v1, vj} in M(G). So propagation can not occur from u1j. Then vj and ui,j and vi are not monitored. Thus in this case, γp(M(G)) =6 1 and it is a contradiction. If vi and vj are adjacen to v1(orv2) in G, then vi, vj and uij cannot monitored in M(G). So γp(M(G)) =6 1 0 and it is a contradiction. Therefore v1v2 is a universal edge and γ (G) = 1.

0 n−3 Theorem 47 ([1]). γ (M(Wn)) = 1 + d 3 e.

12 n−3 Theorem 48. γp(M(Wn)) = 1 + d 3 e.

0 0 n−3 Proof. According to theorem 45 and theorem 47, γp(M(G)) ≤ γ (G) and γ (M(Wn)) = 1 + d 3 e, so n−3 γp(M(Wn)) ≤ 1 + d 3 e. If v1 be a universal vertex of Wn, then it’s clear that to be the smallest size of PD-set, v1 or one of {u12, u13, ··· , u1n} should be in PD-set. Now if v1 be in PD-set, then propagation can not occur from any of {u12, u13, ··· , u1n}, because for example u12 is adjacent to {v2, u2n, u23} and this case can not occur that two of these three vertices are monitored and for monitore the remaining vertex, it is needed that propagation occur from u12. So V (M(Wn)) \{v1, u12, u13, ··· , u1n} induce a n−1 n−3 n−1 M(Cn−1) and we have γp(M(Wn)) = 1+d 3 e but 1+d 3 e ≤ 1+d 3 e and it is a contradiction. Now, suppose that one of {u12, u13, ··· , u1n} be in PD-set. Without losing generality, let u12 be in PD-set. Thus {u13, ··· , u1n}∪{v2, u2n, u23} are monitored in d.s and like previous argument, propagation can not occur from any of {u13, ··· , u1n}. In other hand, V (M(Wn)) \{v1, v2, u2n, u23, u12, u13, ··· , u1n} induce n−3 n−3 a M(Pn−2) and we know γp(M(Pn−2)) = d 3 e, so γp(M(Wn)) = 1 + d 3 e.

0 Theorem 49 ([19]). γ (M(Kh,n−h)) = min{h, n − h}.

Theorem 50. γp(M(Kh,n−h)) = min{h, n − h}.

Proof. Let G = Kh,n−h and V (G) = X ∪ Y such that | X |≤| Y |. If S = X, then S is a PD-set for M(G). So γp(M(G)) ≤| X |. We Know E(G) = {uij | vi and vj are adjacent}, in M(G), induce Kh2Kn−h(cartesian product of Kh and Kn−h) and in [22] proved that γp(Kh2Kn−h) = min{h, n−h}−1. Given the position of Kh2Kn−h in M(Kh,n−h), it is clear that γp(M(Kh,n−h)) ≥ min{h, n − h} − 1. Without losing generality, let min{h, n − h} = h. So if γp-set of M(G) be h − 1 vertices of X or h − 1 vertices of Y , then it is clear that propagation can not occur. Also if V (Kh) = {vi1 , ··· , vih }, then {v1, ··· , vh−1} is a γp-set for Kh2Kn−h but this set can not be a γp-set for M(G) because propagation can not continue. So γp(M(Kh,n−h)) =6 h − 1 and therefore γp(M(Kh,n−h)) = min{h, n − h}.

At the end of this section, we present the following conjecture as an open problem.

0 Conjecture 51. For any graph G, γp(M(G)) = γ (G). Acknowledgments. The authors would like to thank the referee for his/her careful reading and valuable comments which improved the quality of the manuscript.

References

[1] S. Alikhani, N. Soltani N: On the Saturation Number of Graphs. Iranian Journal of Mathematical Chemistry, 9 (4) (2018), 289–299. [2] F. Barioli, W. Barrett, S. Butler, S. M. Cioaba, D. Cvetkovic, S. M. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovic, H. van der Holst, K. Vander Meulen, A. Wangsness (AIM Minimum Rank – Special GraphsWork Group). Forcing sets and the minimum rank of graphs. Linear Algebra and its Applications, 428(7) (2008), 1628–1648. [3] T.L. Baldwin, L. Mili, M.B. Boisen Jr., R. Adapa: Power system observability with minimal phasor measurement placement. IEEE Transactions on Power Systems, 82 (2) (1993), 707–715. [4] F. Barioli, W. Barrett, S. M. Fallat, H. T. Hall, L. Hogben, B. Shader, P. van den Driessche, H. van der Holst: Zero forcing parameters and minimum rank problems. Linear Algebra and its Applications, 433 (1) (2010), 401–411. [5] K. F. Benson, D. Ferrero, M. Flagg, V. Furst, L. Hogben, V. Vasilevska: Nord- haus–Gaddum problems for power domination. Discrete Applied Mathematics, 251 (2018), 103–113.

13 [6] D. Burgarth, V. Giovannetti: Full control by locally induced relaxation.. Physical Review Letters, 99 (2007), 100501. [7] G. Chartrand, L. Lesniak and P. Zhang: Graphs and Digraphs, (5th edition). CRC Press, Boca Raton, Florida, 2011. [8] R. Davila, T. Kalinowsky, S. Stephen: A lower bound on the zero forcing number. Discrete Applied Mathematics, 250 (2018), 363–367. [9] P. Dorbec, M. Mollard, S. Klavzar, S. Spacapan: Power Domination in Product Graphs. SIAM Journal on Discrete Mathematics, 22 (2) (2008), 554–567. [10] P. Dorbec, S. Varghese, A. Vijayakumar: Heredity for generalized power domination. Discrete Mathematics and Theoretical Computer Science, 18 (3) (2016), 11pp. [11] L. Eroh, X. Kang, E. Yi: A Comparison between the Metric Dimension and Zero Forcing Number of Trees and Unicyclic Graphs. Acta Mathematica Sinica(English Series), 33 (6) (2017), 731–747. [12] D. Ferrero, L. Hogben, F. H. J. Kenter, M. Young: Note on power propagation time and lower bounds for the power domination number. Journal of Combinatorial Optimization, 34 (3) (2017), 736–741. [13] G. Garza, N. Shinkel: Which graphs have planar shadow graphs? Pi Mu Epsilon Journal, 11(1) (1999), 11-20. [14] M. Gentner, L. Penso, D. Rautenbach, U. Souza: Extremal values and bounds for the zero forcing number. Discrete Applied Mathematics, 214 (2016), 196–200. [15] M. Gentner, D. Rautenbach: Some bounds on the zero forcing number of a graph. Discrete Applied Mathematics, 236 (2018), 203–213. [16] J. Guo, R. Niedermeier, D. Raible: Improved Algorithms and Complexity Results for Power Domination in Graphs. Algorithmica, 52 (2008), 177–202. [17] T. W. Haynes, S. M. Hedetniemi, S. T. Hedetniemi, M. A. Henning: Domination in graphs applied to electric power networks. SIAM Journal on Discrete Mathematics, 15 (4) (2002), 519–529. [18] L. H. Huang, G. J. Chang, H. G. Yeh: On minimum rank and zero forcing sets of a graph. Linear Algebra and its Applications, 432 (11) (2010), 2961–2973. [19] S. R. Jayaram: Line domination in graphs. Graphs Combinatorics, 3 (1987), 357–363. [20] T. Kalinowsky, N. Kamcev, B. Sudakov: The zero forcing number of graphs. SIAM Journal on Discrete Mathematics, 33 (1) (2019), 95–115. [21] D. D. Row: A technique for computing the zero forcing number of a gragh with a cut-vertex. Linear Algebra and its Applications 436 (2012), 4423–4432. [22] Soh KW, Koh KM: A note on power domination problem in diameter two graphs. AKCE Inter- national Journal of Graphs and Combinatorics, 11(1) (2014), 41–53. [23] B.Thenmozhi, R. Prabha: Power domination of middle graph of path, cycle and star. Interna- tional Journal of Pure and Applied Mathematics, 114(5) (2017), 13–19. [24] S. K. Vaidya, R. M. Pandit: Edge Domination in Some Path and Cycle Related Graphs. ISRN Discrete Mathematics, 2014 (2014), 1–5. [25] S. Varghese: Studies on some generalizations of line graph and the power domination problem in graphs. Ph.D. Thesis, Cochin University of Science and Technology, India; 2011. pp. 153. [26] S. Varghese, A Vijayakumar: On the power domination number of graph products. Algorithms and Discrete Applied Mathematics, 9602 (2016), 357–367.

14 [27] J. V. Vernold: Harmonious coloring of total graphs, n-leaf, central graphs and circumdetic graphs. Ph.D. Thesis, Bharathiar University, Coimbatore, India; 2007. [28] M. Zhao, L. Kang: Power domination in planar graphs with small diameter. Journal of Shanghai University (English Edition), 11 (3) (2007), 218–222. [29] M. Zhao, L. Kang, G. J. Chang: Power domination in graphs. Discrete Mathematics, 306 (2006), 1812–1816.