Perfect Roman domination in trees

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Perfect Roman domination in trees Michael A. Henning a , William F. Klostermeyer b, *, Gary MacGillivray c a b c

Department of Pure and Applied Mathematics, University of Johannesburg, South Africa School of Computing, University of North Florida, Jacksonville, FL 32224-2669, United States Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC Victoria, BC, Canada V8W 3R4

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a b s t r a c t

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Article history: Received 11 January 2017 Received in revised form 19 June 2017 Accepted 30 October 2017 Available online xxxx Keywords: Dominating set Roman dominating function Perfect dominating set Tree

A perfect Roman dominating function on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f (u) = 0 is adjacent to exactly one vertex v for which f (v ) = 2. The weight of a perfect Roman dominating function f is the sum of p the weights of the vertices. The perfect Roman domination number of G, denoted γR (G), is the minimum weight of a perfect Roman dominating function in G. We show that if G is a p tree on n ≥ 3 vertices, then γR (G) ≤ 45 n, and we characterize the trees achieving equality in this bound. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Let G = (V , E) be an undirected graph. Denote the open and closed neighborhoods of a vertex x ∈ V by N(x) and N [x], respectively. That is, N(x) = {v | xv ∈ E } and N [x] = N(x) ∪ {x}. A dominating set of graph G is a set D ⊆ V such that for each u ∈ V \ D, there exists an x ∈ D adjacent to u. The minimum cardinality amongst all dominating sets of G is the domination number, denoted as γ (G). A thorough treatise on dominating sets can be found in [4]. A perfect dominating set is a set S ⊆ V such that for all v ∈ V , |N [v]∩ S | = 1. Perfect dominating sets and several variations on perfect domination have received much attention in the literature; for example, see some discussion in [4] or the survey in [6]. A Roman dominating function of a graph G, abbreviated RD-function, is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f (u) = 0 is adjacent to at least one vertex ∑ v for which f (v ) = 2. The weight of a vertex v is its value, f (v ), assigned to it under f . The weight, w(f ), of f is the sum, u∈V (G) f (u), of the weights of the vertices. The Roman domination number, denoted γR (G), is the minimum weight of an RD-function in G; that is,

γR (G) = min{w(f ) | f is an RD-function in G}. Roman domination was first studied in depth in a graph theory setting in [3], after its initial introduction in the series of papers [8–11]. Roman domination was considered in trees in [5]. In this paper we introduce a perfect version of Roman domination. A perfect Roman dominating function of a graph G, abbreviated PRD-function, is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u with f (u) = 0 is adjacent to exactly one vertex v for which f (v ) = 2. The perfect Roman p

domination number, denoted γR (G), is the minimum weight of a PRD-function in G; that is,

γRp (G) = min{w(f ) | f is a PRD-function in G}.

*

Corresponding author. E-mail addresses: [email protected] (M.A. Henning), [email protected] (W.F. Klostermeyer), [email protected] (G. MacGillivray).

https://doi.org/10.1016/j.dam.2017.10.027 0166-218X/© 2017 Elsevier B.V. All rights reserved.

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Fig. 1. A tree in the family T .

p

p

A PRD-function with minimum weight γR (G) in G is called a γR (G)-function. p Note that every graph with n vertices satisfies γR (G) ≤ n: this can be attained by letting f (v ) = 1 for each vertex in the p

graph. As an example, let P be the Petersen graph. Then γ (P) = 3, γR (P) = 6 and γR (P) = 7 (the latter can be achieved with three vertices of weight 2 and one vertex of weight 1). A different notion of perfection in Roman domination was considered in [7]. In that paper, the authors study Roman dominating functions in which the vertices of weight 1 and 2 induce an independent set. Another related variant of Roman domination in which each vertex of weight 0 must be adjacent to at least two vertices weighted 2 or one vertex weighted 3 is explored in [1]; the vertices with weight 1 must also be adjacent to at least one vertex with weight 2 or 3, though it is shown there that no weight 1 vertices are ever needed. p In this paper, we show that if G is a tree on n ≥ 3 vertices, then γR (G) ≤ 54 n, and we characterize the trees achieving equality in this bound. 2. Notation For a subset S of vertices of a graph G, the subgraph induced by S is denoted by G[S ]. The subgraph obtained from G by deleting all vertices in S and all edges incident with vertices in S is denoted by G − S. The distance between two vertices u and v is the length of a shortest (u, v )-path in G. The eccentricity of a graph G is the maximum distance between any two vertices in G. A leaf is a vertex of degree 1, while its neighbor is a support vertex. A star is the graph K1,k , where k ≥ 1. For a star with k > 1 leaves, the central vertex is the unique vertex of degree greater than one. For r , s ≥ 1, a double star S(r , s) is the tree with exactly two vertices that are not leaves, one of which has r leaf neighbors and the other s leaf neighbors. We denote a path on n vertices by Pn . A rooted tree T distinguishes one vertex r called the root. For each vertex v ̸ = r of T , the parent of v is the neighbor of v on the unique (r , v )-path, while a child of v is any other neighbor of v . The set of children of v is denoted by C (v ). A descendant of v is a vertex u ̸ = v such that the unique (r , u)-path contains v , while an ancestor of v is a vertex u ̸ = v that belongs to the (r , v )-path in T . In particular, every child of v is a descendant of v while the parent of v is an ancestor of v . The grandparent of v is the ancestor of v at distance 2 from v . A grandchild of v is the descendant of v at distance 2 from v . We let D(v ) denote the set of descendants of v , and we define D[v] = D(v ) ∪ {v}. The maximal subtree at v is the subtree of T induced by D[v], and is denoted by Tv . The distance d(u, v ) between two vertices u and v in a connected graph G is the length of a shortest (u, v )-path in G. The maximum distance among all pairs of vertices of G is the diameter of G, denoted by diam(G). The center of a graph G is the set of all vertices of minimum eccentricity, and a central vertex of G is a vertex that belongs to its center. That is, a central vertex of G is a vertex with eccentricity equal to its radius. In particular, a path Pn has a unique central vertex if n is odd, and has two (adjacent) central vertices if n is even. As a shorthand, we shall use the standard notation [k] = {1, . . . , k}. 3. Main result Let T be the family of all trees T whose vertex set can be partitioned into sets, each set inducing a path P5 on five vertices, such that the subgraph induced by the central vertices of these P5 ’s is connected. We call the subtree induced by these central vertices the underlying subtree of the resulting tree T , and we call each such path P5 a base path of the tree T . A tree in the family T with six base paths and whose underlying subtree is a path P6 is illustrated in Fig. 1. We shall prove the following result. p

Theorem 1. If T is a tree of order n ≥ 3, then γR (T ) ≤

4 n, 5

with equality if and only if T ∈ T .

As an immediate corollary of Theorem 1, we have the following result due to Chambers, Kinnersley, Prince and West [2]. Corollary 1 ([2]). If T is a tree of order n ≥ 3, then γR (T ) ≤

4 n, 5

with equality if and only if T ∈ T .

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4. Proof of Theorem 1 In this section, we present a proof of Theorem 1. We first establish useful properties of trees in the family T . Lemma 2. If T is a tree of order n that belongs to the family T and v is an arbitrary vertex of T , then the following hold. p (a) γR (T ) = 45 n. p

(b) There exists a γR (T )-function that assigns to v the value 0 or 1. p R (T )-function

(c) There exists a γ

that assigns to v the value 1 or 2.

Proof. Let T ∈ T have order n, and let the underlying subtree of T have order k, and so n = 5k where k ≥ 1. Let f be an arbitrary PRD-function of T . Let P : uvw xy be an arbitrary base path in T . Thus, v and x are vertices of degree 2 in T that have a common neighbor w , and u and y are the leaf neighbors of v and x, respectively. We note that the vertex w belongs to the underlying subtree of T . The sum of the weights given to u and v by f must be at least 2, unless the weight assigned to w by f is 2. This implies that the sum of the weights assigned by f to the vertices of the base path P is at least 4. Since there are k such (vertex-disjoint) base paths in T , each of which receives a total weight of at least 4 under f , the weight of f is w(f ) ≥ 4k. p Since f is an arbitrary PRD-function of T , this implies that γR (T ) ≥ 4k = 45 n. ∗ Conversely, the function f that assigns the weight 2 to every vertex in the underlying subtree of T , the weight 1 to every p leaf of T , and the weight 0 to every support vertex of T is a PRD-function of T of weight 4k, and so γR (T ) ≤ w(f ∗ ) = 4k = 54 n. p 4 Consequently, γR (T ) = 5 n. This proves Part (a). The function f ∗ defined earlier is a minimum PRD-function of T . Further, we note that the function f ∗∗ that assigns the weight 0 to every vertex in the underlying subtree of T , the weight 1 to exactly one leaf from every base path, the weight 1 to the support neighbor of such a leaf, the weight 0 to the remaining leaves of T , and the weight 2 to the support neighbor of each leaf with weight 0 is a minimum PRD-function of T . Part (b) and Part (c) follow from the existence of these two γRp (T )-functions f ∗ and f ∗∗ . □ We are now in a position to prove Theorem 1. Recall its statement. p

Theorem 1. If T is a tree of order n ≥ 3, then γR (T ) ≤

4 n, 5

with equality if and only if T ∈ T . p

Proof of Theorem 1. We proceed by induction on the order n ≥ 3 of a tree T . If n = 3, then T ∼ = P3 and γR (T ) = 2 < 54 n. This p ′ ′ ′ establishes the base case. Let n ≥ 4 and assume that if T is a tree of order n , where n < n and n′ ≥ 3, then γR (T ′ ) ≤ 45 n′ , ′ with equality if and only if T ∈ T . Let T be a tree of order n. If T is a star, then the function that assigns the weight 2 to the central vertex and the weight 0 to every leaf of the star is p a PRD-function of T of weight 2, and so γR (T ) ≤ 2 < 45 n. Hence, we may assume that diam(T ) ≥ 3. Suppose that diam(T ) = 3, and so T is a double star T ∼ = S(r , s), where r ≥ s ≥ 1. Let u and v be the two vertices of T that are not leaves, where u has r leaf neighbors and v has s leaf neighbors. The function that assigns the weight 2 to the vertex u, the weight 1 to the leaf neighbors of v , and the weight 0 to the remaining vertices of T is a PRD-function of T of weight 2 + s, p and so γR (T ) ≤ 2 + s ≤ 23 (s + 1) = 34 (2s + 2) ≤ 34 (r + s + 2) = 43 n < 45 n. Hence, we may assume that diam(T ) ≥ 4, for otherwise the desired result follows. Let u and r be two vertices at maximum distance apart in T . Necessarily, u and r are leaves and d(u, r) = diam(T ). We now root the tree T at the vertex r. Let v be the parent of u, w the parent of v , x the parent of w , and y the parent of x. We note that if diam(T ) = 4, then y = r; otherwise, y ̸ = r. The remainder of the proof proceeds by establishing eight claims and then deducing from those claims that the statement of the theorem is true. p

Claim 1. If dT (v ) ≥ 4, then γR (T ) <

4 n. 5

Proof. Suppose that dT (v ) ≥ 4. Let T ′ be the tree obtained from T by deleting v and its children; that is, T ′ = T − V (Tv ) where recall that Tv denotes the maximal subtree of T at v induced by D[v]. Let T ′ have order n′ , and so n′ = n − dT (v ) ≤ n − 4. p Since diam(T ) ≥ 4, we note that n′ ≥ 3. Applying the inductive hypothesis to the tree T ′ , γR (T ′ ) ≤ 45 n′ ≤ 45 (n − 4). Let f ′ be p ′ ′ ′ a γR (T )-function. If f (w ) ∈ {1, 2}, then we can extend f to a PRD-function f of T by assigning the weight 2 to the vertex v and the weight 0 to the children of v . In this case, the resulting function f has weight w(f ) = w(f ′ ) + 2. If f ′ (w ) = 0, then let f be the function obtained from f ′ by re-assigning to w the weight 1 and leaving the weight of all other vertices under f ′ unchanged, and by assigning the weight 2 to the vertex v and the weight 0 to the children of v . In this case, the resulting p function f has weight w(f ) = w(f ′ ) + 3. In both cases, γR (T ) ≤ w(f ) ≤ w(f ′ ) + 3 ≤ 54 (n − 4) + 3 < 45 n. □ By Claim 1, we may assume that every child of w in T has degree at most 3, for otherwise the desired result follows. For i ∈ [3], let ti be the number of children of w of degree i. In particular, t1 is the number of leaf neighbors of w . Further, since v has degree 2 or 3, we note that t2 + t3 ≥ 1. By assumption, dT (w) = t1 + t2 + t3 + 1. Please cite this article in press as: M.A. Henning, et al., Perfect Roman domination in trees, Discrete Applied Mathematics (2017), https://doi.org/10.1016/j.dam.2017.10.027.

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Claim 2. If t3 ≥ 1 and (t1 , t2 , t3 ) ̸ ∈ {(0, 1, 1), (0, 2, 2)}, then γR (T ) <

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4 n. 5

Proof. Suppose that t3 ≥ 1 and (t1 , t2 , t3 ) ̸ ∈ {(0, 1, 1), (0, 2, 2)}. Let T ′ be the tree obtained from T by deleting the t3 children of w of degree 3 and all their leaf neighbors. Let T ′ have order n′ , and note that n′ ≥ 3 and n′ = n − 3t3 . Applying the inductive p p hypothesis to the tree T ′ , γR (T ′ ) ≤ 45 n′ = 54 (n − 3t3 ). Among all γR (T ′ )-functions, let f ′ be chosen so that the weight assigned to w is as large as possible. If f ′ (w ) ∈ {1, 2}, then we can extend f ′ to a PRD-function f of T by assigning the weight 2 to each child of w of degree 3 and the weight 0 to the leaf neighbors of each such child. The resulting function f is a PRD-function of T of p weight w(f ) = w(f ′ ) + 2t3 . Thus, γR (T ) ≤ w(f ) = w(f ′ ) + 2t3 ≤ 45 (n − 3t3 ) + 2t3 = 54 n − 52 t3 < 54 n, noting that t3 ≥ 1. Hence, we may assume that f ′ (w ) = 0, for otherwise the desired bound follows. Since f ′ (w ) = 0, no leaf neighbor of w has weight 0 under f ′ . Further, if a leaf neighbor of w has weight 2 under f ′ , then p re-assigning to both this leaf neighbor and to the vertex w the weight 1 produces a new γR (T ′ )-function that assigns to w a ′ larger weight than assigned to it under f , a contradiction. Hence, every leaf neighbor of w has weight 1 under f ′ . p

Claim 2.1. If t3 ≥ 3, then γR (T ) <

4 n. 5

Proof. Suppose that t3 ≥ 3. Let f be the function obtained from f ′ by re-assigning to w the weight 1 and leaving the weight of all other vertices under f ′ unchanged, and by assigning the weight 2 to each child of w of degree 3 and the weight 0 to the leaf neighbors of each such child. The resulting function f is a PRD-function of T of weight w(f ) = w(f ′ ) + 1 + 2t3 . Thus, γRp (T ) ≤ w(f ) = w(f ′ ) + 1 + 2t3 ≤ 54 (n − 3t3 ) + 1 + 2t3 ≤ 54 n − 52 t3 + 1 < 54 n, as t3 ≥ 3. □ By Claim 2.1, we may assume that t3 ≤ 2, for otherwise the desired result follows. p

Claim 2.2. If t3 = 2, then γR (T ) <

4 n. 5

Proof. Suppose that t3 = 2. Let T ′′ be the tree obtained from T by deleting w and all its descendants; that is, T ′′ = T − V (Tw ). Let T ′′ have order n′′ . We note that {x, y} ⊆ V (T ′′ ), and so n′′ ≥ 2. Suppose that n′′ = 2. In this case, the tree T is determined and n = |{w, x, y}| + t1 + 2t2 + 3t3 = 3 + t1 + 2t2 + 3t3 . Assigning the weight 2 to w , the weight 0 to all neighbors of w (including its parent x), and the weight 1 to the remaining vertices of T ′′ (at distance 2 from w ) produces a PRD-function of T of weight 2 + t2 + 2t3 ≤ 32 (3 + t1 + 2t2 + 3t3 ) = 23 n < 54 n. Hence, we may assume that n′′ ≥ 3, for otherwise the desired upper bound follows. Applying the inductive hypothesis to p p p the tree T ′′ , we have that γR (T ′′ ) ≤ 45 n′′ = 54 (n − 1 − t1 − 2t2 − 3t3 ). Let f ′′ be a γR (T ′′ )-function, and so w(f ′′ ) = γR (T ′′ ). If t1 + t2 = 0, then we can extend f ′′ to a PRD-function f of T by assigning the weight 1 to w , the weight 2 to the two children of w (of degree 3) and the weight 0 to the four grandchildren of w . The resulting PRD-function f has

+ 5 = 45 (n − 7) + 5 < 45 n. Hence, we may assume that t1 + t2 ≥ 1. If f ′′ (x) ∈ {1, 2}, then we can extend f ′′ to a PRD-function f of T by assigning the weight 2 to w , the weight 0 to all children of w , and the weight 1 to the remaining vertices of T . The resulting PRD-function f has weight w(f ) = w(f ′′ ) + 2 + t2 + 2t3 ≤ 4 ′′ n + 2 + t2 + 2t3 = 45 (n − 1 − t1 − 2t2 − 3t3 ) + 2 + t2 + 2t3 = 54 n + 65 − 54 t1 − 53 t2 − 25 t3 . Since t1 + t2 ≥ 1 and t3 = 2, 5 this implies that w(f ) < 54 n. Hence, we may assume that f ′′ (x) = 0. weight w(f ) = w(f ′′ ) + 5 ≤

4 ′′ n 5

Let f be the function obtained from f ′′ by re-assigning to x the weight 1 and leaving the weight of all other vertices under f unchanged, and by assigning the weight 2 to w , the weight 0 to all children of w , and the weight 1 to the remaining vertices of T . Noting that t3 = 2, the resulting PRD-function f has weight w(f ) = w(f ′′ ) + 3 + t2 + 2t3 ≤ 54 n′′ + 3 + t2 + 2t3 = 4 n + 11 − 45 t1 − 35 t2 − 52 t3 = 45 n + 51 (7 − 4t1 − 3t2 ). If t1 ≥ 2 or if t1 + t2 ≥ 3, this implies that w(f ) < 45 n. Hence, we may 5 5 ′′

assume that t1 ≤ 1 and t1 + t2 ≤ 2. Suppose that t1 = 1. Since t1 + t2 ≤ 2, we note that t2 ≤ 1. If t2 = 1, then 7 − 4t1 − 3t2 = 0, implying that w(f ) ≤ 45 n. Suppose that in this case, w(f ) = 45 n. In particular, this implies that w(f ′′ ) = 54 n′′ and, by the inductive hypothesis, T ′′ ∈ T . p By Lemma 2(c), we could have chosen the γR (T ′′ )-function f ′′ so that f ′′ (x) ∈ {1, 2}, a contradiction to our assumption that ′′ f (x) = 0. Therefore, t2 ̸ = 1, implying that t2 = 0. We can now extend f ′′ to a PRD-function f of T by assigning the weight 1 to w , the weight 2 to the two children of w of degree 3, the weight 1 to the leaf neighbor of w , and the weight 0 to the four grandchildren of w . The resulting PRD-function f has weight w(f ) = w(f ′′ ) + 6 ≤ 54 n′′ + 6 = 54 (n − 8) + 6 < 54 n. Hence, we may assume that t1 = 0. By assumption, t1 + t2 ≥ 1 and t1 + t2 ≤ 2, implying that t2 ∈ {1, 2}. If t2 = 2, then (t1 , t2 , t3 ) = (0, 2, 2), contradicting our supposition that (t1 , t2 , t3 ) ̸ = (0, 2, 2). Hence, t2 = 1. We can now extend f ′′ to a PRD-function f of T by assigning the weight 2 to the two children of w of degree 3, the weight 0 to the four grandchildren of w whose parents have degree 3, and the weight 1 to the remaining three vertices of T . The resulting PRD-function f has weight w(f ) = w(f ′′ ) + 7 ≤ 4 (n − 9) + 7 < 45 n. □ 5

4 ′′ n 5

+7=

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We continue with the proof of Claim 2. By Claim 2.2, we may assume that t3 = 1, for otherwise the desired result follows. Renaming vertices if necessary, we may assume that v is the child of w of degree 3. Thus, T ′ = T − V (Tv ). Recall that f ′ is a γRp (T ′ )-function. Further, f ′ (w) = 0 and every leaf neighbor of w has weight 1 under f ′ . Suppose that a child v ′ of w of degree 2 has weight 2 under f ′ . Thus, v ′ is the only neighbor of w in T ′ that has weight 2 under f ′ . Let u′ be the leaf neighbor of v ′ . By the minimality of f ′ , we note that f ′ (u′ ) = 0. Let f be the function obtained from f ′ by re-assigning to u′ and v ′ the weight 1, and leaving the weight of all other vertices under f ′ unchanged, and by assigning the weight 2 to the vertex v and the weight 0 to its two children. The resulting function f has weight w(f ) = w(f ′ ) + 2. p

− 3) + 2 < 54 n. Hence, we may assume that every child of w of degree 2, if any, has weight 1 under f ′ , for otherwise the desired bound follows. Thus, every child of w in T ′ has weight 1 under f ′ , implying that every grandchild of w in T ′ if it exists (such a grandchild of w is a leaf whose parent has degree 2) has weight 1 under f ′ . With these assumptions, the only neighbor of w of weight 2 under f ′ is the parent of w , namely the vertex x; that is, f ′ (w ) = 0 and f ′ (x) = 2. Suppose next that t1 + t2 ≥ 2. In this case, let f be the function obtained from f ′ by re-assigning to w the weight 2, re-assigning to every child of w in T ′ the weight 0, leaving the weight of all other vertices under f ′ unchanged, and by assigning the weight 2 to the vertex v and the weight 0 to its two leaf neighbors. The resulting function f has weight w(f ) = w(f ′ ) + 4 − t1 − t2 ≤ w(f ′ ) + 2 ≤ 45 (n − 3) + 2 < 54 n. Hence, we may assume that t1 + t2 ≤ 1, for otherwise the

Thus, γR (T ) ≤ w(f ) = w(f ′ ) + 2 ≤

4 (n 5

desired bound follows. Suppose that t1 = 1, implying that t2 = 0. Let T ′′ be the tree constructed in the proof of Claim 2.2. Thus, T ′′ = T − V (Tw ) is obtained from T by deleting w and all its descendants. Following the notation in the proof of Claim 2.2, let T ′′ have order n′′ . p Since here (t1 , t2 , t3 ) = (1, 0, 1), we note that n′′ = n − 5. If n′′ = 2, then, as before, γR (T ) < 45 n. Hence, we may assume p 4 ′′ ′′ ′′ ′′ ′′ that n ≥ 3. Let f be a γR (T )-function. By the inductive hypothesis, w(f ) ≤ 5 n = 54 (n − 5). We now extend f ′′ to a PRD-function f of T by assigning the weight 1 to w and its leaf neighbor, the weight 2 to v , and the weight 0 to its two children. The resulting PRD-function f has weight w(f ) = w(f ′′ ) + 4 ≤ 45 (n − 5) + 5 = 54 n. Suppose that in this case when t1 = 1, that w(f ) = 45 n. This implies w(f ′′ ) = 54 n′′ and, by the inductive hypothesis, p that T ′′ ∈ T . By Lemma 2(c), we could have chosen the γR (T ′′ )-function f ′′ so that f ′′ (x) ∈ {0, 1}. With this choice of p ′′ ′′ our γR (T )-function, f can be extended to a PRD-function f ∗ of T by assigning the weight 2 to v , the weight 0 to the neighbors of v (including its parent w ), and the weight 0 to the leaf neighbor of w . The resulting PRD-function f ∗ has weight w(f ∗ ) = w(f ′′ ) + 3 = 54 (n − 5) + 3 < 54 n. Hence, we may assume that t1 = 0. If t2 = 1, then (t1 , t2 , t3 ) = (0, 1, 1), a contradiction. Hence, t2 = 0. We once again consider the tree T ′′ = T − V (Tw ) p which in this case has order n′′ = n − 4. Let f ′′ be a γR (T ′′ )-function. By the inductive hypothesis, w(f ′′ ) ≤ 54 n′′ = 45 (n − 4). ′′ We now extend f to a PRD-function f of T by assigning the weight 1 to w , the weight 2 to v , and the weight 0 to the two children of v . The resulting PRD-function f has weight w(f ) = w(f ′′ ) + 3 ≤ 54 (n − 4) + 3 < 45 n. This completes the proof of Claim 2. □ p

Claim 3. If t3 = 0 and (t1 , t2 ) ̸ ∈ {(0, 3), (1, 2)}, then γR (T ) ≤ p (t1 , t2 ) = (0, 2) and γR (T ) = 54 n, then T ∈ T .

4 n, 5

with strict inequality if (t1 , t2 ) ̸ = (0, 2). Further, if

Proof. Suppose that t3 = 0. Thus, every child of w is a leaf or a support vertex of degree 2. In particular, dT (v ) = 2, implying that t2 ≥ 1. Let T ′ be the tree obtained from T by deleting w and all its descendants; that is, T ′ = T − V (Tw ). Let T ′ have order n′ . We note that n′ = n − 1 − t1 − 2t2 . Further, since {x, y} ⊆ V (T ′ ), we have that n′ ≥ 2. Suppose that n′ = 2. In this case, the tree T is determined and n = 3 + t1 + 2t3 . Assigning the weight 2 to w , the weight 0 to all neighbors of w (including its parent x), and the weight 1 to the remaining vertices of T produces a PRD-function f of T of weight w(f ) = 3 + t2 . However, 54 n = 45 (3 + t1 + 2t3 ) = (3 + t2 ) + 15 (4t1 + 3t2 − 3) ≥ 3 + t2 , as t1 ≥ 0 and t2 ≥ 1. Further, if w(f ) = 45 n, then we must have equality throughout this inequality chain, implying that t1 = 0 and t2 = 1 and therefore that T ∼ = P5 ∈ T . Hence, we may assume that n′ ≥ 3, for otherwise the desired upper bound follows. Applying the inductive p p p hypothesis to the tree T ′ , we have that γR (T ′ ) ≤ 54 n′ = 45 (n − 1 − t1 − 2t2 ). Let f ′ be a γR (T ′ )-function, and so w(f ′ ) = γR (T ′ ). ′ ′ Suppose that t1 ≥ 2 or t2 ≥ 4. If f (x) ∈ {1, 2}, then we extend f to a PRD-function f of T by assigning the weight 2 to w , the weight 0 to all children of w , and the weight 1 to the grandchildren of w . The resulting PRD-function f of T has weight w(f ) = w(f ′ ) + 2 + t2 ≤ 54 (n − 1 − t1 − 2t2 ) + 2 + t2 = 54 n + 65 − 45 t1 − 35 t2 < 54 n, noting that t1 ≥ 2 or t2 ≥ 4. Hence, we may assume that f ′ (x) = 0. Let f be the function obtained from f ′′ by re-assigning to x the weight 1 and leaving the weight of all other vertices under f ′′ unchanged, and by assigning the weight 2 to w , the weight 0 to all children of w , and the weight 1 to the grandchildren of w . The resulting PRD-function f has weight w(f ) = w(f ′′ ) + 3 + t2 ≤ 45 (n − 1 − t1 − 2t2 ) + 3 + t2 = 4 n + 15 (11 − 4t1 − 3t2 ) ≤ 45 n, noting that t2 ≥ 1 and t1 ≥ 2 or t2 ≥ 4. Further, if w(f ) = 45 n, then we must have equality 5 throughout this inequality chain, implying that t1 = 2 and t2 = 1. Further, w(f ′ ) = 45 n′ and, by the inductive hypothesis, p T ′ ∈ T . By Lemma 2(c), we could have chosen the γR (T ′ )-function f ′ so that f ′ (x) ∈ {1, 2}, contrary to assumption. Hence, 4 w(f ) < 5 n, and we may therefore assume that t1 ≤ 1 and t2 ≤ 3. Suppose that t1 = 1. If t2 = 3, then, letting f be the function defined in the preceding paragraph, we note that w(f ) ≤ w(f ′ ) + 3 + t2 ≤ 54 n + 15 (11 − 4t1 − 3t2 ) < 54 n. Hence, we may assume that in this case when t1 = 1 that t2 = 1 or t2 = 2. If t2 = 2, then (t1 , t2 ) = (1, 2), a contradiction. Hence, t2 = 1. If f ′ (x) ∈ {1, 2}, then we extend f ′ to a PRD-function f of T by assigning the weight 2 to w , the weight 0 to both children of w , and the weight 1 to the grandchild of w . If f ′ (x) = 0, Please cite this article in press as: M.A. Henning, et al., Perfect Roman domination in trees, Discrete Applied Mathematics (2017), https://doi.org/10.1016/j.dam.2017.10.027.

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(a) T1 .

(b) T2 .

(c) T3 .

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(d) T4 .

Fig. 2. The four level-2 subtrees of T .

then we extend f ′ to a PRD-function f of T by assigning the weight 2 to v , the weight 0 to w and u, and the weight 1 to the child of w that is a leaf. In both cases, the resulting PRD-function f of T has weight w(f ) = w(f ′ ) + 3 ≤ 45 (n − 4) + 3 < 54 n. Hence, we may assume that t1 = 0, for otherwise the desired upper bound follows. Recall that t2 ≤ 3. If t2 = 3, then (t1 , t2 ) = (0, 3), a contradiction. Hence, t2 = 1 or t2 = 2. Suppose that t2 = 1. In this case, n′ = n − 3. If f ′ (x) ∈ {0, 1}, then we extend f ′ to a PRD-function f of T by assigning the weight 2 to v , and the weight 0 to both u and w . If f ′ (x) = 2, then we extend f ′ to a PRD-function f of T by assigning the weight 0 to w , and the weight 1 to both u and v . In both cases, the resulting PRD-function f of T has weight w(f ) = w(f ′ ) + 2 ≤ 45 (n − 3) + 2 < 54 n. Hence, we may assume that t2 = 2. Since t1 = t3 = 0 and t2 = 2, we note that n′ = n − 5. If f ′ (x) ∈ {1, 2}, then we extend f ′ to a PRD-function f of T by assigning the weight 2 to w , the weight 0 to both children of w , and the weight 1 to both grandchildren of w . If f ′ (x) = 0, then we extend f ′ to a PRD-function f of T by assigning the weight 2 to v , the weight 0 to u and w , and the weight 1 to the two descendants of w different from u and v . In both cases, the resulting PRD-function f of T has weight w(f ) = w(f ′ ) + 4 ≤ 54 (n − 5) + 4 ≤ 54 n. Further, suppose that w(f ) = 45 n. This implies that w(f ′ ) = 45 n′ and that, by the inductive hypothesis, T ′ ∈ T . Let P : abcde be the base path in T ′ that contains the vertex y (so y is one of the vertices {a, b, c , d, e}), where the vertex c belongs to the underlying subtree of T ′ ∈ T . Suppose that y is a leaf of P. Renaming vertices of P if necessary, we may assume that y = a. Let fa be the function that assigns the weight 2 to the vertex w and to every vertex in the underlying subtree of T ′ , the weight 1 to every leaf of T different from a, and the weight 0 to the leaf a and to every support vertex of T . The resulting function fa is a PRD-function p p of T of weight w(fa ) = (γR (T ′ ) − 1) + 4 = 54 n′ + 3 = 45 n − 1, implying that γR (T ) ≤ w(fa ) < 54 n. Suppose that y is a support vertex of P. Renaming vertices of P if necessary, we may assume that y = b. Let fb be the function that assigns the weight 0 to every vertex in the underlying subtree of T ′ , the weight 1 to the leaf a and to exactly one leaf from every other base path, the weight 0 to the remaining leaves of T ′ (including the leaf e) and to the vertex b, the weight 1 to every support vertex different from b that has a leaf neighbor of weight 1, the weight 2 to the support vertex that has a leaf neighbor of weight 0, the weight 2 to the vertex w , the weight 0 to both children of w , and the weight 1 to both p grandchildren of w . The resulting function fb is a PRD-function of T of weight w(fb ) = (γR (T ′ ) − 1) + 4 = 45 n − 1, implying p 4 that γR (T ) ≤ w(fb ) < 5 n. p Hence, we may assume that y is the central vertex, c, of P, for otherwise γR (T ) < 54 n. In this case, T ∈ T , where the path Tw ∼ = P5 with w as its central vertex is a base path of T and where the underlying subtree of T is formed from the underlying subtree of T ′ by adding to it the vertex w and the edge w y. This completes the proof of Claim 3. □ By Claims 2 and 3, we may assume that (t1 , t2 , t3 ) ∈ {(0, 1, 1), (1, 2, 0), (0, 3, 0), (0, 2, 2)}, for otherwise the desired result follows. Thus, the maximal subtree, Tw , of T rooted at w is isomorphic to one of the trees T1 , T2 , T3 and T4 shown in Fig. 2. We note that the tree T1 is associated with the triple (t1 , t2 , t3 ) = (0, 1, 1), the tree T2 with the triple (t1 , t2 , t3 ) = (1, 2, 0), the tree T3 with the triple (t1 , t2 , t3 ) = (0, 3, 0), and the tree T4 with the triple (t1 , t2 , t3 ) = (0, 2, 2). We call the subtree Tw a level-2 subtree of T . Further, if Tw ∼ = Ti for some i ∈ [4], we call the subtree Tw a level-2 subtree of Type-i. More generally, if w ′ is a vertex in T at distance diam(T ) − 2 from the root r such that w ′ has a grandchild (at distance diam(T ) from r), then we call the maximal subtree of T rooted at a vertex w ′ a level-2 subtree of T . As shown with the vertex w , we may assume that every level-2 subtree of T is a level-2 subtree of Type-i for some i ∈ [4], for otherwise the desired result follows. We now consider the parent, x, of w in the rooted tree T . Let s1 be the number of children of x that are leaves, and let s2 be the number of children of x of degree 2 that have a leaf neighbor. Further, let s3 be the number of children of x of degree at least 3 all of whose children are leaves. We note that a child of x that has no grandchild does not belong to a level-2 subtree of T . We now consider the structure of the subtree Tx rooted at x. p

Claim 4. If s3 ≥ 1, then γR (T ) <

4 n. 5

Proof. Suppose that s3 ≥ 1. Thus, x has a child, w ′ , of degree at least 3 where every child of w ′ is a leaf (and so in this case, w′ ̸= w). Let T ′ be obtained from T by deleting w and all descendants of w , and by deleting w′ and all children of w′ ; that is, T ′ = T − V (Tw ) − V (Tw′ ). Let T ′ have order n′ . We note that {x, y} ⊆ V (T ′ ). If n′ = 2, then the tree T ′ is determined. In this p case, a simple case analysis of the four possibilities for the level-2 subtree Tw shows that in all cases γR (T ) < 54 n. Hence, we Please cite this article in press as: M.A. Henning, et al., Perfect Roman domination in trees, Discrete Applied Mathematics (2017), https://doi.org/10.1016/j.dam.2017.10.027.

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may assume that n′ ≥ 3. We note that n′ = n − |V (Tw )| − |V (Tw′ )| ≤ n − |V (Tw )| − 3. Applying the inductive hypothesis to p p the tree T ′ , we have γR (T ′ ) ≤ 54 n′ . Among all γR (T ′ )-functions, let f ′ be chosen so that the weight assigned to x is as large as possible. If f ′ (x) ∈ {1, 2}, then we extend f ′ to a PRD-function f of T by assigning the weight 2 to both w and w ′ , the weight 0 to all children of w and w ′ , and the weight 1 to the grandchildren of w . If Tw ∼ = T1 , then n′ ≤ n − 9 and w(f ) = w(f ′ ) + 7 ≤ 4 4 4 ′ ′ ∼ n + 7 ≤ 5 (n − 9) + 7 < 5 n. If Tw = T2 , then n ≤ n − 9 and w(f ) = w(f ′ ) + 6 ≤ 54 n′ + 6 ≤ 45 (n − 9) + 6 < 45 n. If 5 Tw ∼ = T3 , then n′ ≤ n − 10 and w(f ) = w(f ′ ) + 7 ≤ 54 n′ + 7 ≤ 54 (n − 10) + 7 < 45 n. If Tw ∼ = T4 , then n′ ≤ n − 14 and p 4 ′ 4 4 4 ′ w(f ) = w(f ) + 10 ≤ 5 n + 10 ≤ 5 (n − 14) + 10 < 5 n. In all cases, w(f ) < 5 n, implying that γR (T ) ≤ w (f ) < 54 n. Hence, p p we may assume that f ′ (x) = 0. This implies, by our choice of the γR (T ′ )-function f ′ , that every γR (T ′ )-function assigns to x the weight 0. Suppose that Tw ∼ = T1 , where v is the child of w of degree 3 and v ′ is the child of w of degree 2. Let f be the function obtained from f ′ by re-assigning to x the weight 1 and leaving the weight of all other vertices under f ′ unchanged, and by assigning the weight 2 to v and w ′ , the weight 1 to v ′ and its child, and the weight 0 to the remaining vertices. The resulting PRD-function f of T has weight w(f ) = w(f ′ ) + 7 ≤ 45 (n − 9) + 7 < 54 n. Suppose that Tw ∼ = Ti for some i ∈ {2, 3, 4}. Let f be the function obtained from f ′ by re-assigning to x the weight 1 and leaving the weight of all other vertices under f ′ unchanged, and by assigning the weight 2 to both w and w ′ , the weight 0 to all children of w and w ′ , and the weight 1 to the grandchildren of w . If Tw ∼ = T2 , then n′ ≤ n − 9 and w(f ) = w(f ′ ) + 7 ≤ 4 4 4 ′ ′ ∼ n + 7 ≤ 5 (n − 9) + 7 < 5 n. If Tw = T3 , then n ≤ n − 10 and w(f ) = w(f ′ ) + 8 ≤ 54 n′ + 8 ≤ 45 (n − 10) + 8 = 45 n. 5 If Tw ∼ = T4 , then n′ ≤ n − 14 and w(f ) = w(f ′ ) + 11 ≤ 45 n′ + 11 ≤ 54 (n − 14) + 11 < 54 n. In all cases, w(f ) ≤ 54 n, with strict inequality if Tw ∼ = T2 or Tw ∼ = T4 . Moreover, suppose that w(f ) = 54 n. In this case, Tw ∼ = T3 and dT (w′ ) = 3. Further, p 4 ′ ′ w(f ) = 5 n , implying by the inductive hypothesis that T ′ ∈ T . By Lemma 2(c), we could have chosen the γR (T ′ )-function p ′ ′ ′ f so that f (x) ∈ {1, 2}, contrary to our assumption that every γR (T )-function assigns to x the weight 0. Hence, if Tw ∼ = T3 , then w(f ) < 45 n. This completes the proof of Claim 4. □ By Claim 4, we may assume that s3 = 0, for otherwise the desired result holds. Thus, every child of x of degree at least 3 belongs to a level-2 subtree of T . Hence if w ′ is a child of x of degree at least 3, then w ′ has at least one grandchild, implying by our earlier assumptions that Tw′ is a level-2 subtree of Type-i for some i ∈ [4]. p

Claim 5. If T has a level-2 subtree of Type-1, then γR (T ) <

4 n. 5

Proof. Suppose that T has a level-2 subtree of Type-1. Renaming vertices if necessary, we may assume that Tw is a level-2 subtree of Type-1; that is, Tw ∼ = T1 . Further, we may assume that v is the child of w of degree 3 and v ′ is the child of w of ′ degree 2. Let u be the child of v ′ . We note that u′ is a leaf. p

Claim 5.1. If s1 ≥ 1, then γR (T ) <

4 n. 5

Proof. Suppose that s1 ≥ 1. Thus, a child of x, say w ′ , is a leaf in T . We now consider the tree T ′ = T − (V (Tw ) ∪ {w ′ }). Let T ′ have order n′ , and so n′ = n − 7. We note that {x, y} ⊆ V (T ′ ). If n′ = 2, then the tree T ′ is determined and n = 9. In this case, assigning the weight 2 to both x and w , the weight 1 to each of the three grandchildren of w , and the weight 0 to the remaining vertices of T produces a PRD-function of weight 7 < 36 = 54 n. Hence, we may assume that n′ ≥ 3. 5 p p 4 ′ 4 ′ ′ Applying the inductive hypothesis to the tree T , γR (T ) ≤ 5 n = 5 (n − 7). Let f ′ be a γR (T ′ )-function. If f ′ (x) = 2, then ′ ′ we extend f to a PRD-function f of T by assigning the weight 0 to both w and w , and the weight 1 to all five descendants of w . If f ′ (x) ∈ {0, 1}, then we extend f ′ to a PRD-function f of T by assigning the weight 2 to v , the weight 0 to the three neighbors of v (including its parent w ), and the weight 1 to each of u′ , v ′ and w ′ . In both cases, the PRD-function f has weight w(f ) = w(f ′ ) + 5 ≤ 45 (n − 7) + 5 < 45 n. □ p

Claim 5.2. If s2 ≥ 1, then γR (T ) <

4 n. 5

Proof. Suppose that s2 ≥ 1. Thus, x has a child, w ∗ , of degree 2 whose child, v ∗ , is a leaf. Let T ′ = T − V (Tw ) − {v ∗ , w ∗ } and let T ′ have order n′ . We note that n′ = n − 8 and that {x, y} ⊆ V (T ′ ). If n′ = 2, then the tree T ′ is determined and n = 10. In this case, assigning the weight 2 to both v and w ∗ , the weight 1 to the three vertices u′ , v ′ and y, and the weight 0 to the remaining vertices of T produces a PRD-function of T of weight 7 < 54 n. Hence, we may assume that n′ ≥ 3. Applying the p p inductive hypothesis to the tree T ′ , γR (T ′ ) ≤ 54 n′ = 45 (n − 8). Let f ′ be a γR (T ′ )-function. If f ′ (x) = 2, then we extend f ′ to a ∗ PRD-function f of T by assigning the weight 0 to both w and w , and the weight 1 to all six descendants of w . If f ′ (x) ∈ {0, 1}, then we extend f ′ to a PRD-function f of T by assigning the weight 2 to v , the weight 0 to the three neighbors of v , and the weight 1 to each of u′ , v ′ , v ∗ and w ∗ . In both cases, the PRD-function f has weight w(f ) = w(f ′ ) + 6 ≤ 54 (n − 8) + 6 < 45 n. □ p

Claim 5.3. If some child of x belongs to a level-2 subtree of Type-2, then γR (T ) <

4 n. 5

Please cite this article in press as: M.A. Henning, et al., Perfect Roman domination in trees, Discrete Applied Mathematics (2017), https://doi.org/10.1016/j.dam.2017.10.027.

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Proof. Suppose that T has a level-2 subtree Tw∗ of Type-2 for some child w ∗ of x (different from w ). Thus, Tw∗ ∼ = T2 . Let T ′ = T − V (Tw ) − V (Tw∗ ) and let T ′ have order n′ . We note that n′ = n − 12 and that {x, y} ⊆ V (T ′ ). If n′ = 2, then the tree T ′ is determined and n = 14. In this case, assigning the weight 2 to both v and w ∗ , the weight 0 to all neighbors of v and w ∗ , and the weight 1 to the remaining five vertices of T produces a PRD-function of weight 9 < 45 · 14 = 54 n. Hence, we may p p assume that n′ ≥ 3. Applying the inductive hypothesis to the tree T ′ , γR (T ′ ) ≤ 45 n′ = 45 (n − 12). Let f ′ be a γR (T ′ )-function. ′ ′ ∗ If f (x) ∈ {1, 2}, then we extend f to a PRD-function f of T by assigning the weight 2 to both v and w , the weight 0 to all children of v and w ∗ , and the weight 1 to the remaining five vertices of V (T ) \ V (T ′ ) (including the vertex w ). If f ′ (x) = 0, then we extend f ′ to a PRD-function f of T by assigning the weight 2 to v , the weight 2 to exactly one child, say v ∗ , of w ∗ of degree 2, the weight 0 to the neighbors of v and v ∗ , and the weight 1 to the remaining five vertices of V (T ) \ V (T ′ ). In both cases, the PRD-function f has weight w(f ) = w(f ′ ) + 9 ≤ 45 (n − 12) + 9 < 54 n. □ p

Claim 5.4. If some child of x belongs to a level-2 subtree of Type-3, then γR (T ) <

4 n. 5

Proof. Suppose that T has a level-3 subtree Tw∗ of Type-2 for some child w ∗ of x (different from w ). Thus, Tw∗ ∼ = T3 . Let T ′ = T − V (Tw ) − V (Tw∗ ) and let T ′ have order n′ . We note that n′ = n − 13 and that {x, y} ⊆ V (T ′ ). If n′ = 2, then the tree T ′ is determined and n = 15. In this case, assigning the weight 2 to both v and w ∗ , the weight 0 to all neighbors of v and w ∗ , and the weight 1 to the remaining six vertices of T produces a PRD-function of weight 10 < 45 · 15 = 54 n. Hence, we may p p assume that n′ ≥ 3. Applying the inductive hypothesis to the tree T ′ , γR (T ′ ) ≤ 45 n′ = 45 (n − 13). Let f ′ be a γR (T ′ )-function. If f ′ (x) ∈ {1, 2}, then we extend f ′ to a PRD-function f of T by assigning the weight 2 to both v and w ∗ , the weight 0 to all children of v and w ∗ , and the weight 1 to the remaining vertices of V (T ) \ V (T ′ ) (including the vertex w ). If f ′ (x) = 0, then we extend f ′ to a PRD-function f of T by assigning the weight 2 to v , the weight 2 to exactly one child, say v ∗ , of w ∗ of degree 2, the weight 0 to the neighbors of v and v ∗ , and the weight 1 to the remaining vertices of V (T ) \ V (T ′ ). In both cases, the PRD-function f has weight w(f ) = w(f ′ ) + 10 ≤ 54 (n − 13) + 10 < 54 n. □ p

Claim 5.5. If some child of x belongs to a level-2 subtree of Type-4, then γR (T ) <

4 n. 5

Proof. Suppose that T has a level-4 subtree Tw∗ of Type-2 for some child w ∗ of x (different from w ). Thus, Tw∗ ∼ = T4 . Let T ′ = T − V (Tw ) − V (Tw∗ ) and let T ′ have order n′ . We note that n′ = n − 17 and that {x, y} ⊆ V (T ′ ). If n′ = 2, then the tree T ′ is determined and n = 19. In this case, assigning the weight 2 to both v and w ∗ , the weight 0 to all neighbors of v and w ∗ , and the weight 1 to the remaining nine vertices of T produces a PRD-function of weight 13 < 54 · 19 = 54 n. p Hence, we may assume that n′ ≥ 3. Applying the inductive hypothesis to the tree T ′ , γR (T ′ ) ≤ 45 n′ = 54 (n − 17). Let p f ′ be a γR (T ′ )-function. If f ′ (x) ∈ {1, 2}, then we extend f ′ to a PRD-function f of T by assigning the weight 2 to both v and w ∗ , the weight 0 to all children of v and w ∗ , and the weight 1 to the remaining vertices of V (T ) \ V (T ′ ) (including the vertex w ). If f ′ (x) = 0, then we extend f ′ to a PRD-function f of T by assigning the weight 2 to v , the weight 2 to both children of w ∗ of degree 3, the weight 0 to the neighbors of v , the weight 0 to the four grandchildren of w ∗ that have a parent of degree 3, and the weight 1 to the remaining seven vertices of V (T ) \ V (T ′ ). In both cases, the PRD-function f has weight w(f ) = w(f ′ ) + 13 ≤ 45 (n − 17) + 13 < 45 n. □ We now return to the proof of Claim 5. By Claims 5.1, 5.2, 5.3, 5.4 and 5.5, we may assume that every child of x belongs to a p level-2 subtree of Type-1, for otherwise γR (T ) < 45 n. We now consider the tree T ′ obtained from T by deleting all descendants ′ ′ ′ of x. Let T have order n . We note that n = n − 6(dT (x) − 1) = n − 6dT (x) + 6, since we remove from T the dT (x) − 1 level-2 subtrees of Type-1 associated with the children of x, each of which has order 6. We also note that {x, y} ⊆ V (T ′ ), and so n′ ≥ 2. If n′ = 2, then the tree T ′ is determined and n = 6dT (x) − 4. In this case, assigning the weight 2 to all grandchildren of x of degree 3, the weight 0 to all neighbors of such grandchildren, and the weight 1 to the remaining vertices of T (including x and y) produces a PRD-function f of weight w(f ) = 2 + 4(dT (x) − 1) = 4dT (x) − 2. However, since dT (x) ≥ 2, we note that 4 n = 45 (6dT (x) − 4) = (4dT (x) − 2) + 54 (dT (x) − 32 ) > 4dT (x) − 2, implying that w(f ) < 54 n. Hence, we may assume that 5 p p ′ n ≥ 3. Applying the inductive hypothesis to the tree T ′ , γR (T ′ ) ≤ 45 n′ = 45 (n − 6dT (x) + 6). Let f ′ be a γR (T ′ )-function. ′ ′ ′ By construction and our earlier assumptions, x is a leaf in T . If f (x) = 2, then the minimality of f implies that f ′ (y) = 0. In this case, we can simply re-assign to both x and y the weight 1, and leave the weight of all other vertices under f ′ unchanged p to produce a new γR (T ′ )-function. Hence, we may choose f ′ so that f ′ (x) ∈ {0, 1}. We now extend f ′ to a PRD-function f of T by assigning the weight 2 to all grandchildren of x of degree 3, the weight 0 to all neighbors of such grandchildren, and the weight 1 to the remaining vertices of V (T ) \ V (T ′ ). The resulting PRD-function f has weight w(f ) = w(f ′ ) + 4(dT (x) − 1) ≤ 4 (n − 6dT (x) + 6) + 4(dT (x) − 1) = 45 n − 54 (dT (x) − 1) < 54 n. This completes the proof of Claim 5. □ 5 By Claim 5, we may assume that there is no level-2 subtree of Type-1 in T , for otherwise the desired result follows. In particular, Tw is a level-2 subtree of T of Type-2, Type-3 or Type-4. Let s4 be the number of children of x that belong to a level-2 subtree of T . We note that at least one child of x, namely the vertex w , belongs to a level-2 subtree of T , and so s4 ≥ 1. p

Claim 6. If s4 ≥ 2, then γR (T ) <

4 n. 5

Please cite this article in press as: M.A. Henning, et al., Perfect Roman domination in trees, Discrete Applied Mathematics (2017), https://doi.org/10.1016/j.dam.2017.10.027.

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Proof. Suppose that s4 ≥ 2. For i ∈ {2, 3, 4}, let s4,i be the number of children of x that belong to a level-2 subtree of T of Type-i. Thus, s4 = s4,2 + s4,3 + s4,4 . We note that each level-2 subtree of T of Type-2, Type-3 and Type-4 contains six, seven and eleven vertices, respectively. We now consider the tree T ′ obtained from T by deleting all descendants of x that belong to a level-2 subtree. Let T ′ have order n′ . We note that n′ = n − 6s4,2 − 7s4,3 − 11s4,4 , since s4,i level-2 subtrees are deleted from T to produce T ′ , for each i ∈ {2, 3, 4}. We also note that {x, y} ⊆ V (T ′ ) and so n′ ≥ 2. If n′ = 2, then the tree T ′ is determined and n = 2 + 6s4,2 + 7s4,3 + 11s4,4 . In this case, assigning the weight 2 to every child of x, the weight 0 to every grandchild of x, and the weight 1 to the remaining vertices of T (including x and y) produces a PRD-function f with weight w(f ) = 2 + 4s4,2 + 5s4,3 + 8s4,4 . Since s4 ≥ 2, we note that 4 5

n =

4 5

(2 + 6s4,2 + 7s4,3 + 11s4,4 ) 1

= (2 + 4s4,2 + 5s4,3 + 8s4,4 ) + (4s4,2 + 3s4,3 + 4s4,4 − 2) 5

1

≥ w(f ) + (3s4,2 + 3s4,3 + 3s4,4 − 2) 5 1

≥ w(f ) + (3s4 − 2) 5 > w(f ). p

Hence, we may assume that n′ ≥ 3. Applying the inductive hypothesis to the tree T ′ , γR (T ′ ) ≤ 45 n′ = 45 (n − 6s4,2 − 7s4,3 − p 11s4,4 ). Let f ′ be a γR (T ′ )-function. If f ′ (x) ∈ {1, 2}, then we extend f ′ to a PRD-function f of T by assigning the weight 2 to every child of x, the weight 0 to every grandchild of x, and the weight 1 to the remaining vertices of T . If f ′ (x) = 0, then let f be the function obtained from f ′ by re-assigning to x the weight 1 and leaving the weight of all other vertices under f ′ unchanged, and by assigning the weight 2 to every child of x, the weight 0 to every grandchild of x, and the weight 1 to the remaining vertices of T . The resulting PRD-function f has weight w(f ) ≤ (w(f ′ ) + 1) + 4s4,2 + 5s4,3 + 8s4,4 . Since s4 ≥ 2, we note that w(f ) ≤ (w(f ′ ) + 1) + 4s4,2 + 5s4,3 + 8s4,4

≤ = = ≤ = ≤ <

4 5 4 5 4 5 4 5 4 5 4 5 4

n′ + 1 + 4s4,2 + 5s4,3 + 8s4,4 (n − 6s4,2 − 7s4,3 − 11s4,4 ) + 1 + 4s4,2 + 5s4,3 + 8s4,4 n+ n+ n+ n−

1 5 1 5 1 5 1

(5 − 4s4,2 − 3s4,3 − 4s4,4 ) (5 − 3s4,2 − 3s4,3 − 3s4,4 ) (5 − 3s4 )

5

n. 5 This completes the proof of Claim 6.

□

By Claim 6, we may assume that s4 ≤ 1, for otherwise the desired result follows. As observed earlier, s4 ≥ 1. Consequently, s4 = 1. p

Claim 7. If s1 + s2 = 0, then γR (T ) <

4 n. 5

Proof. Suppose that s1 + s2 = 0. Since s3 = 0 (by Claim 4) and s4 = 1, this implies that dT (x) = 2. Thus, w is the only child of x in T . Let T ′ be the tree obtained from T by deleting x and all its descendants; that is, T ′ = T − V (Tx ). Let T ′ have order n′ . If s4 = s4,2 , then n′ = n − 7. If s4 = s4,3 , then n′ = n − 8. If s4 = s4,4 , then n′ = n − 12. We note that y ∈ V (T ′ ), and so n′ ≥ 1. Suppose that n′ ∈ {1, 2}, and so the tree T is determined. In this case, we assign the weight 2 to w , the weight 0 to every neighbor of w (including its parent x), and the weight 1 to the remaining vertices of T . Let f be the resulting PRD-function of T . If n′ = 1 and s4 = s4,2 , then n = 8 and w(f ) = 5. If n′ = 1 and s4 = s4,3 , then n = 9 and w(f ) = 6. If n′ = 1 and s4 = s4,4 , then n = 13 and w(f ) = 9. If n′ = 2 and s4 = s4,2 , then n = 9 and w(f ) = 6. If n′ = 2 and s4 = s4,3 , then n = 10 and w(f ) = 7. If n′ = 2 and s4 = s4,4 , then n = 14 and w(f ) = 10. In all cases, w(f ) < 54 n. Hence, we may assume that n′ ≥ 3. p p Applying the inductive hypothesis to the tree T ′ , γR (T ′ ) ≤ 45 n′ . Let f ′ be a γR (T ′ )-function. ′ We now extend f to a PRD-function f of T by assigning the weight 2 to w , the weight 0 to every neighbor of w (including its parent x), and the weight 1 to the remaining vertices of V (T ) \ V (T ′ ). If s4 = s4,2 , then w(f ) = w(f ′ ) + 5 ≤ 45 n′ + 5 = 4 (n − 7) + 5 < 45 n. If s4 = s4,3 , then w(f ) = w(f ′ ) + 6 ≤ 54 n′ + 6 = 45 (n − 8) + 6 < 54 n. If s4 = s4,4 , then 5 w(f ) = w(f ′ ) + 9 ≤ 54 n′ + 9 = 45 (n − 12) + 9 < 45 n. In all three cases, w(f ) < 54 n. This completes the proof of Claim 7. □ Please cite this article in press as: M.A. Henning, et al., Perfect Roman domination in trees, Discrete Applied Mathematics (2017), https://doi.org/10.1016/j.dam.2017.10.027.

10

M.A. Henning et al. / Discrete Applied Mathematics ( p

Claim 8. If s1 + s2 ≥ 2, then γR (T ) <

)

–

4 n. 5

Proof. Suppose that s1 + s2 ≥ 2. Let T ′ be the tree obtained from T by deleting w and all its descendants; that is, T ′ = T − V (Tw ). Let T ′ have order n′ . If s4 = s4,2 , then n′ = n − 6. If s4 = s4,3 , then n′ = n − 7. If s4 = s4,4 , then p p n′ = n − 11. We note that n′ ≥ 4. Applying the inductive hypothesis to the tree T ′ , γR (T ′ ) ≤ 45 n′ . Let f ′ be a γR (T ′ )-function. ′ ′ If f (x) ∈ {1, 2}, then we can extend f to a PRD-function f of T by assigning the weight 2 to w , the weight 0 to every child of w , and the weight 1 to every grandchild of w . If s4 = s4,2 , then w(f ) = w(f ′ ) + 4 ≤ 45 n′ + 4 = 45 (n − 6) + 4 < 45 n. If s4 = s4,3 , then w(f ) = w(f ′ ) + 5 ≤ 45 n′ + 5 = 45 (n − 7) + 5 < 45 n. If s4 = s4,4 , then w(f ) = w(f ′ ) + 8 ≤ 54 n′ + 8 = 45 (n − 11) + 8 < 54 n. In all three cases, w(f ) < 45 n. Hence, we may assume that f ′ (x) = 0, for otherwise the desired result follows. Suppose that f ′ (y) ∈ {0, 1}. In this case, all children of x have weight 1 under f ′ , except for exactly one child of x, say w ′ , which has degree 2 and has weight 2 under f ′ . Let v ′ be the child of w ′ , and note that f ′ (v ′ ) = 0. We now simply re-assign to both v ′ and w ′ the weight 1, leaving the weight of all other vertices under f ′ unchanged, and extended the resulting function to a PRD-function f of T by assigning the weight 2 to w , the weight 0 to every child of w , and the weight 1 to every grandchild of w . If s4 = s4,2 , then w(f ) = w(f ′ ) + 4. If s4 = s4,3 , then w(f ) = w(f ′ ) + 5. If s4 = s4,4 , then w(f ) = w(f ′ ) + 8. In all three cases, proceeding exactly as in the previous paragraph, w(f ) < 45 n. Hence, we may assume that f ′ (y) = 2, for otherwise the desired result follows. This implies that every child and every grandchild of x in T ′ has weight 1 under f ′ . We now let f be the PRD-function of T obtained from f ′ by re-assigning to x the weight 2, re-assigning to all children of x in T ′ the weight 0, leaving the weight of all other vertices under f ′ unchanged, and assigning the weight 2 to w , the weight 0 to every child of w , and the weight 1 to every grandchild of w . By supposition, s1 + s2 ≥ 2. If s4 = s4,2 , then w(f ) = w(f ′ ) + (2 − s1 − s2 ) + 4 ≤ 45 n′ + 4 = 45 (n − 6) + 4 < 54 n. If s4 = s4,3 , then w(f ) = w(f ′ ) + (2 − s1 − s2 ) + 5 ≤ 4 ′ n + 5 = 45 (n − 7) + 5 < 54 n. If s4 = s4,4 , then w(f ) = w(f ′ ) + (2 − s1 − s2 ) + 8 ≤ 54 n′ + 8 = 54 (n − 11) + 8 < 45 n. In all 5 three cases, we get that w(f ) < 45 n. This completes the proof of Claim 8. □ We are now ready to complete the proof. By Claims 7 and 8, we may assume that s1 + s2 = 1, for otherwise the desired result follows. Thus, either s1 = 0 and s2 = 1 or s1 = 1 and s2 = 2. Recall that s3 = 0 and s4 = 1. In what follows, let T ′ be the tree obtained from T by deleting x and all its descendants; that is, T ′ = T − V (Tx ). Let T ′ have order n′ . We examine the following possibilities: If s1 = 1 and s4 = s4,2 , then n′ = n − 8. If s2 = 1 and s4 = s4,2 , then n′ = n − 9. If s1 = 1 and s4 = s4,3 , then n′ = n − 9. If s2 = 1 and s4 = s4,3 , then n′ = n − 10. If s1 = 1 and s4 = s4,4 , then n′ = n − 13. If s2 = 1 and s4 = s4,4 , then n′ = n − 14. We note that since y ∈ V (T ′ ), it follows that n′ ≥ 1. Suppose that n′ ∈ {1, 2}, and so the tree T is determined. In this case, we assign the weight 2 to x and w , the weight 0 to every neighbor of x different from w , the weight 0 to every child of w , and the weight 1 to the remaining vertices of T . Let f be the resulting PRD-function of T . We examine the following possibilities: If n′ If n′ If n′ If n′ If n′ If n′

= 1, and s4 = s4,2 , then n = 9 and w(f ) = 6. If n′ = 1, s2 = 1, and s4 = s4,2 , then n = 10 and w(f ) = 7. = 1, and s4 = s4,3 , then n = 10 and w(f ) = 7. If n′ = 1, s2 = 1, and s4 = s4,3 , then n = 11 and w(f ) = 8. = 1, and s4 = s4,4 , then n = 14 and w(f ) = 10. If n′ = 1, s2 = 1, and s4 = s4,4 , then n = 15 and w(f ) = 11. = 1, and s4 = s4,2 , then n = 10 and w(f ) = 7. If n′ = 2, s2 = 1, and s4 = s4,2 , then n = 11 and w(f ) = 8. = 1, and s4 = s4,3 , then n = 11 and w(f ) = 8. If n′ = 2, s2 = 1, and s4 = s4,3 , then n = 12 and w(f ) = 9. = 1, and s4 = s4,4 , then n = 15 and w(f ) = 11. If n′ = 2, s2 = 1, and s4 = s4,4 , then n = 16 and w(f ) = 12. p In all cases, w(f ) < 54 n. Hence, we may assume that n′ ≥ 3. Applying the inductive hypothesis to the tree T ′ , γR (T ′ ) ≤ 45 n′ . p ′ ′ Let f be a γR (T )-function. We now extend f ′ to a PRD-function f of T by assigning the weight 2 to w , the weight 0 to every child of w , and the weight 1 to the remaining vertices of V (T ) \ V (T ′ ). We examine the following possibilities: If s1 = 1 and s4 = s4,2 , then n′ = n − 8 and w(f ) = w(f ′ ) + 6 ≤ 45 n′ + 6 = 45 (n − 8) + 6 < 54 n. If s2 = 1 and s4 = s4,2 , then n′ = n − 9 and w(f ) = w(f ′ ) + 7 ≤ 45 n′ + 7 = 45 (n − 9) + 7 < 54 n. If s1 = 1 and s4 = s4,3 , then n′ = n − 9 and w(f ) = w(f ′ ) + 7 ≤ 45 n′ + 7 = 45 (n − 9) + 7 < 54 n. If s1 = 1 and s4 = s4,4 , then n′ = n − 13 and w(f ) = w(f ′ ) + 10 ≤ 54 n′ + 10 = 45 (n − 13) + 10 < 54 n. If s2 = 1 and s4 = s4,4 , then n′ = n − 14 and w(f ) = w(f ′ ) + 11 ≤ 54 n′ + 11 = 45 (n − 14) + 11 < 54 n. = 1, s1 = 1, s1 = 1, s1 = 2, s1 = 2, s1 = 2, s1

In all these cases, w(f ) < 45 n. Hence we may assume that none of the above cases occur, for otherwise the desired result follows. Hence, the only case remaining is the case when s2 = 1 and s4 = s4,3 . Since s2 = 1 and s4 = s4,3 , we note that n′ = n − 10 and w(f ) = w(f ′ ) + 8 ≤ 54 n′ + 8 = 45 (n − 10) + 8 = 45 n. Suppose that w(f ) = 45 n. This implies that w(f ′ ) = 54 n′ , and therefore by the inductive hypothesis that T ′ ∈ T . By Lemma 2(b), we can p choose the γR (T ′ )-function f ′ so that f ′ (x) ∈ {0, 1}. We can now extend the function f ′ to a PRD-function f of T by assigning the weight 2 to w , the weight 0 to every neighbor of w (including its parent x), and the weight 1 to the remaining vertices of V (T ) \ V (T ′ ). The resulting PRD-function f has weight w(f ) = w(f ′ ) + 7 = 54 n′ + 7 = 45 (n − 10) + 7 < 45 n. This completes the proof of Theorem 1. □ p

We note that every PRD-function is an RD-function, implying that for every graph G, γR (G) ≤ γR (G). Analogously as in the proof of Lemma 2(a), if T is a tree of order n that belongs to the family T , then γR (T ) = 54 n. Please cite this article in press as: M.A. Henning, et al., Perfect Roman domination in trees, Discrete Applied Mathematics (2017), https://doi.org/10.1016/j.dam.2017.10.027.

M.A. Henning et al. / Discrete Applied Mathematics (

)

(a) G1 .

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11

(b) G2 .

Fig. 3. The graphs G1 and G2 .

5. Closing remarks Our main result (see Theorem 1) establishes a tight upper bound on the perfect Roman domination number of a tree on p at least three vertices in terms of it order. More precisely, we prove that if T is a tree on n ≥ 3 vertices, then γR (T ) ≤ 45 n, and this bound is tight. We remark that adding edges to a tree can possibly increase its perfect Roman domination number. Hence, it is not clear whether our tree result can be extended to all connected graphs. Indeed, there exist graphs with perfect Roman domination number larger than any of its spanning trees, as the following result shows. p

p

Lemma 3. For every k ≥ 1, there exists a connected graph Gk such that γR (Gk ) − γR (T ) ≥ k for every spanning tree T of Gk . Proof. For k ≥ 1, let Gk be the graph obtained from two disjoint stars both isomorphic to K1,k+2 by adding k + 1 new vertices and joining each new vertex to the central vertex of both stars. (The graphs G1 and G2 are illustrated in Fig. 3(a) and (b), p p respectively). It is a simple exercise to check that γR (Gk ) = k + 5 and that γR (T ) = 5 for every spanning tree T of Gk . □ p

p

By Lemma 3, if G is a connected graph, then it not necessarily true that γR (G) ≤ γR (T ) for some spanning tree T of G. Thus, it is not clear how to apply Theorem 1 in order to extend the tree result to all connected graphs. We further note that there exist trees such that adding an edge can decrease the perfect Roman domination number: a star with two or more leaves p p with a pendant vertex attached to one of the leaves is one example. In this example, γR (T ) = 3 and γR (G) = 2, where G is the graph that results from adding an edge from the center of the star to the pendant vertex that was attached to one of the initial leaves. We close with the following problem that we have yet to settle. Problem 1. Determine a tight upper bound on the perfect Roman domination number of a connected graph on at least three p vertices in terms of its order. In particular, is it true that if G is a connected graph of order n ≥ 3, then γR (G) ≤ 45 n? Acknowledgments The first author’s research was supported in part by the South African National Research Foundation and the University of Johannesburg. The third author was supported by the Natural Sciences and Engineering Research Council of Canada. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

R. Beeler, T. Haynes, S.T. Hedetniemi, Double Roman domination, Discrete Appl. Math. 211 (2016) 23–29. E.W. Chambers, B. Kinnersley, N. Prince, D.B. West, Extremal problems for Roman domination, SIAM J. Discrete Math. 23 (2009) 1575–1586. E.J. Cockayne, P.A. Dreyer, S.M. Hedetniemi, S.T. Hedetniemi, Roman domination in graphs, Discrete Math. 278 (1–3) (2004) 11–22. T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998. M. Henning, A characterization of Roman trees, Discuss. Math. Graph Theory 22 (2) (2002) 325–334. W. Klostermeyer, A taxonomy of perfect domination, J. Discrete Math. Sci. Cryptogr. 18 (2015) 105–116. N. Rad, L. Volkmann, Roman domination perfect graphs, An. St. Univ. Ovidius Constanta 19 (2011) 167–174. C.S. ReVelle, Can you protect the Roman Empire? Johns Hopkins Mag. 49 (2) (1997) 40. C.S. ReVelle, Test your solution to Can you protect the Roman Empire? Johns Hopkins Mag. 49 (3) (1997) 70. C.S. ReVelle, K.E. Rosing, Defendens Imperium Romanum: a classical problem in military, Amer. Math. Monthly 107 (7) (2000) 585–594. I. Stewart, Defend the Roman Empire! Sci. Am. (1999) 136–138.

Please cite this article in press as: M.A. Henning, et al., Perfect Roman domination in trees, Discrete Applied Mathematics (2017), https://doi.org/10.1016/j.dam.2017.10.027.