
Citation: Robert Russell Monteith Paterson. Depletion of Indonesian oil palm plantations implied from modeling oil palm mortality and Ganoderma boninense rot under future climate[J]. AIMS Environmental Science, 2020, 7(5): 366-379. doi: 10.3934/environsci.2020024
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In this paper, we shall only consider graphs without multiple edges or loops. Let G=(V(G),E(G)) be a graph, v∈V(G), the neighborhood of v in G is denoted by N(v). That is to say N(v)={u|uv∈E(G),u∈V(G)}. The degree of a vertex v is denoted by d(v), i.e. d(v)=|N(v)|. A graph is trivial if it has a single vertex. The maximum degree and the minimum degree of a graph G are denoted by Δ(G) and δ(G), respectively. Denote by Kn the complete graph on n vertices.
A subset D of the vertex set of a graph G is a dominating set if every vertex not in D has at least one neighbor in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A dominating set D of G with |D|=γ(G) is called a γ-set of G.
Roman domination of graphs is an interesting variety of domination, which was proposed by Cockayne et al. [6]. A Roman dominating function (RDF) of a graph G is a function f:V(G)→{0,1,2} such that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. The weight w(f) of a Roman dominating function f is the value w(f)=∑u∈V(G)f(u). The minimum weight of an RDF on a graph G is called the Roman domination number γR(G) of G. An RDF f of G with w(f)=γR(G) is called a γR-function of G. The problems on domination and Roman domination of graphs have been investigated widely, for example, see list of references [8,9,10,13] and [3,7,12], respectively.
In 2016, Chellali et al. [5] introduced a variant of Roman dominating functions, called Roman {2}-dominating functions. A Roman {2}-dominating function (R{2}DF) of G is a function f:V→{0,1,2} such that ∑u∈N(v)f(u)≥2 for every vertex v∈V with f(v)=0. The weight of a Roman {2}-dominating function f is the sum ∑v∈Vf(v). The Roman {2}-domination number γ{R2}(G) is the minimum weight of an R{2}DF of G. Note that if f is an R{2}DF of G and v is a vertex with f(v)=0, then either there is a vertex u∈N(v) with f(u)=2, or at least two vertices x,y∈N(v) with f(x)=f(y)=1. Hence, an R{2}DF of G is also an RDF of G, which is also mentioned by Chellali et al [5]. Moreover, they showed that the decision problem for Roman {2}-domination is NP-complete, even for bipartite graphs.
In fact, a Roman {2}-dominating function is essentially the same as a weak {2}-dominating function, which was introduced by Brešar et al. [1] and studied in literatures [2,11,14,15].
For a mapping f:V(G)→{0,1,2}, let (V0,V1,V2) be the ordered partition of V(G) induced by f such that Vi={x:f(x)=i} for i=0,1,2. Note that there exists a 1-1 correspondence between the function f and the partition (V0,V1,V2) of V(G), so we will write f=(V0,V1,V2).
Chellali et al. [4] obtained the following lower bound of Roman domination number.
Lemma 1. (Chellali et al. [4]) Let G be a nontrivial connected graph with maximum degree Δ. Then γR(G)≥Δ+1Δγ(G).
In this paper, we generalize this result on nontrivial connected graph G with maximum degree Δ and minimum degree δ. We prove that γR(G)≥Δ+2δΔ+δγ(G). As a corollary, we obtain that 32γ(G)≤γR(G)≤2γ(G) for any nontrivial regular graph G. Moreover, we prove that γR(G)≤2γ{R2}(G)−1 for every graph G and there exists a graph Ik such that γ{R2}(Ik)=k and γR(Ik)=2k−1 for any integer k≥2.
Lemma 2. (Cockayne et al. [6]) Let f=(V0,V1,V2) be a γR-function of an isolate-free graph G with |V1| as small as possible. Then
(i) No edge of G joins V1 and V2;
(ii) V1 is independent, namely no edge of G joins two vertices in V1;
(iii) Each vertex of V0 is adjacent to at most one vertex of V1.
Theorem 3. Let G be a nontrivial connected graph with maximum degree Δ(G)=Δ and minimum degree δ(G)=δ. Then
γR(G)≥Δ+2δΔ+δγ(G). | (2.1) |
Moreover, if the equality holds, then
γ(G)=n(Δ+δ)Δδ+Δ+δandγR(G)=n(Δ+2δ)Δδ+Δ+δ. |
Proof. Let f=(V0,V1,V2) be a γR-function of G with V1 as small as possible. By Lemma 2, we know that N(v)⊆V0 for any v∈V1 and N(v1)∩N(v2)=∅ for any v1,v2∈V1. So we have
|V1|≤|V0|δ | (2.2) |
Since G is nontrivial, it follows that V2≠∅. Note that every vertex in V2 is adjacent to at most Δ vertices in V0; we have
|V0|≤Δ|V2| | (2.3) |
By Formulae (2.2) and (2.3), we have
|V1|≤Δδ|V2| | (2.4) |
By the definition of an RDF, every vertex in V0 has at least one neighbor in V2. So V1∪V2 is a dominating set of G. Together with Formula (2.4), we can obtain that
γ(G)≤|V1|+|V2|≤Δδ|V2|+|V2|=Δ+δδ|V2|. |
Note that f is a γR-function of G; we have
γR(G)=|V1|+2|V2|=(|V1|+|V2|)+|V2|≥γ(G)+δΔ+δγ(G)=Δ+2δΔ+δγ(G). |
Moreover, if the equality in Formula (2.1) holds, then by previous argument we obtain that |V1|=|V0|δ, |V0|=Δ|V2|, and V1∪V2 is a γ-set of G. Then we have
n=|V0|+|V1|+|V2|=|V0|+|V0|δ+|V0|Δ=Δδ+Δ+δΔδ|V0|. |
Hence, we have
|V0|=nΔδΔδ+Δ+δ,|V1|=nΔΔδ+Δ+δ, and |V2|=nδΔδ+Δ+δ. |
So
γR(G)=|V1|+2|V2|=n(Δ+2δ)Δδ+Δ+δ and γ(G)=|V1|+|V2|=n(Δ+δ)Δδ+Δ+δ |
since V1∪V2 is a γ-set of G. This completes the proof.
Now we show that the lower bound in Theorem 3 can be attained by constructing an infinite family of graphs. For any integers k≥2, δ≥2 and Δ=kδ, we construct a graph Hk from K1,Δ by adding k news vertices such that each new vertex is adjacent to δ vertices of K1,Δ with degree 1 and no two new vertices has common neighbors. Then add some edges between the neighbors of each new vertex u such that δ(Hk)=δ and the induced subetaaph of N(u) in Hk is not complete. The resulting graph Hk is a connected graph with maximum degree Δ(G)=Δ and maximum degree δ(G)=δ. It can be checked that γ(Hk)=k+1 and γR(Hk)=k+2=Δ+2δΔ+δγ(G).
For example, if k=2, δ=3 and Δ=kδ=6, then the graph H2 constructed by the above method is shown in Figure 1, where u1 and u2 are new vertices.
Furthermore, by Theorem 3, we can obtain a lower bound of the Roman domination number on regular graphs.
Corollary 4. Let G be an r-regular graph, where r≥1. Then
γR(G)≥32γ(G) | (2.5) |
Moreover, if the equality holds, then
γ(G)=2nr+2andγR(G)=3nr+2. |
Proof. Since G is r-regular, we have Δ(G)=δ(G)=r. By Theorem 3 we can obtain that this corollary is true.
For any integer n≥2, denote by G2n the (2n−2)-regular graph with 2n vertices, namely G2n is the graph obtained from K2n by deleting a perfect matching. It can be checked that γ(G2n)=2 and γR(G2n)=3=32γ(G) for any n≥2. Hence, the bound in Corollary 4 is attained.
Note that γR(G)≤2γ(G) for any graph G; we can conclude the following result.
Corollary 5. Let G be an r-regular graph, where r≥1. Then
32γ(G)≤γR(G)≤2γ(G). |
Chellali et al. [5] obtain the following bounds for the Roman {2}-domination number of a graph G.
Lemma 6. (Chellali et al. [5]) For every graph G, γ(G)≤γ{R2}(G)≤γR(G)≤2γ(G).
Lemma 7. (Chellali et al. [5]) If G is a connected graph of order n and maximum degree Δ(G)=Δ, then
γ{R2}(G)≥2nΔ+2. |
Theorem 8. For every graph G, γR(G)≤2γ{R2}(G)−1. Moreover, for any integer k≥2, there exists a graph Ik such that γ{R2}(Ik)=k and γR(Ik)=2k−1.
Proof. Let f=(V0,V1,V2) be an γ{R2}-function of G. Then γ{R2}(G)=|V1|+2|V2| and γR(G)≤2|V1|+2|V2| since V1∪V2 is a dominating set of G. If |V2|≥1, then γR(G)≤2|V1|+2|V2|=2γ{R2}(G)−2|V2|≥2γ{R2}(G)−2. If |V2|=0, then every vertex in V0 is adjacent to at least two vertices in V1. So for any vertex u∈V1, f′=(V0,{u},V1∖{u}) is an RDF of G. Then we have γR(G)≤1+2|V1∖{u}|=2|V1|−1=2γ{R2}(G)−1.
For any integer k≥2, let Ik be the graph obtained from Kk by replacing every edge of Kk with two paths of length 2. Then Δ(Ik)=2(k−1) and δ(Ik)=2. We first prove that γ{R2}(Ik)=k. Since V(Ik)=|V(Kk)|+2|E(Kk)|=k+2⋅k(k−1)2=k2, by Lemma 7 we can obtain γ{R2}(Ik)≥2|V(Ik)|Δ(Ik)+2=2k22(k−1)+2=k. On the other hand, let f(x)=1 for each x∈V(Ik) with d(x)=2(k−1) and f(y)=0 for each y∈V(Ik) with d(y)=2. It can be seen that f is an R{2}DF of Ik and w(f)=k. Hence, γ{R2}(Ik)=k.
We now prove that γR(Ik)=2k−1. Let g={V′1,V′2,V′3} be a γR-function of Ik such that |V′1| is minimum. For each 4-cycle C=v1v2v3v4v1 of Ik with d(v1)=d(v3)=2(k−1) and d(v2)=d(v4)=2, we have wg(C)=g(v1)+g(v2)+g(v3)+g(v4)≥2. If wg(C)=2, then by Lemma 2(iii) we have g(vi)∈{0,2} for any i∈{1,2,3,4}. Hence, one of v1 and v3 has value 2 and g(v2)=g(v4)=0. If wg(C)=3, then by Lemma 2(i) we have {g(v1),g(v3)}={1,2} or {g(v2),g(v4)}={1,2}. When {g(v2),g(v4)}={1,2}, let {g′(v1),g′(v2)}={1,2}, g′(v2)=g′(v4)=0 and g′(x)=g(x) for any x∈V(Ik)∖{v1,v2,v3,v4}. Then g′ is also a γR-function of Ik. If wg(C)=4, then exchange the values on C such that v1,v3 have value 2 and v2,v4 have value 0. So we obtain that Ik has a γR-function h such that h(y)=0 for any y∈V(Ik) with degree 2. Note that any two vertices of Ik with degree 2(k−1) belongs to a 4-cycle considered above; we can obtain that there is exactly one vertex z of Ik with degree 2(k−1) such that h(z)=1. Hence, γR(Ik)=w(h)=2k−1.
Note that the graph Ik constructed in Theorem 8 satisfies that γ(Ik)=k=γ{R2}(Ik). By Theorem 8, it suffices to prove that γ(Ik)=k. Let A={v:v∈V(Ik),d(v)=2(k−1)} and B=V(Ik)∖A. We will prove that Ik has a γ-set containing no vertex of B. Let D be a γ-set of Ik. If D contains a vertex u∈B. Since the degree of u is 2, let u1 and u2 be two neighbors of u in Ik. Then d(u1)=d(u2)=2(k−1) and, by the construction of Ik, u1 and u2 have two common neighbors u,u′ with degree 2. Hence, at least one of u′,u1, and u2 belongs to D. Let D′=(D∖{u,u′})∪{u1,u2}. Then D′ is also a γ-set of Ik. Hence, we can obtain a γ-set of Ik containing no vertex of B by performing the above operation for each vertex v∈D∩B. So A is a γ-set of Ik and γ(Ik)=|A|=k.
By Lemma 6 and Theorem 8, we can obtain the following corollary.
Corollary 9. For every graph G, γ{R2}(G)≤γR(G)≤2γ{R2}(G)−1.
Theorem 10. For every graph G, γR(G)≤γ(G)+γ{R2}(G)−1.
Proof. By Lemma 6 we can obtain that γR(G)≤2γ(G)≤γ(G)+γ{R2}(G). If the equality holds, then γR(G)=2γ(G) and γ(G)=γ{R2}(G). So γR(G)=2γ{R2}(G), which contradicts Theorem 8. Hence, we have γR(G)≤γ(G)+γ{R2}(G)−1.
In this paper, we prove that γR(G)≥Δ+2δΔ+δγ(G) for any nontrivial connected graph G with maximum degree Δ and minimum degree δ, which improves a result obtained by Chellali et al. [4]. As a corollary, we obtain that 32γ(G)≤γR(G)≤2γ(G) for any nontrivial regular graph G. Moreover, we prove that γR(G)≤2γ{R2}(G)−1 for every graph G and the bound is achieved. Although the bounds in Theorem 3 and Theorem 8 are achieved, characterizing the graphs that satisfy the equalities remain a challenge for further work.
The author thanks anonymous referees sincerely for their helpful suggestions to improve this work. This work was supported by the National Natural Science Foundation of China (No.61802158) and Natural Science Foundation of Gansu Province (20JR10RA605).
The author declares that they have no conflict of interest.
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