Algorithmic aspects of Roman domination in graphs

For a simple, undirected graph G=(V,E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G = (V, E)$$\end{document}, a Roman dominating function (RDF) f:V→{0,1,2}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f{:}V \rightarrow \lbrace 0, 1, 2 \rbrace $$\end{document} has the property that, every vertex u with f(u)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(u) = 0$$\end{document} is adjacent to at least one vertex v for which f(v)=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(v) = 2$$\end{document}. The weight of a RDF is the sum f(V)=∑v∈Vf(v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(V) = \sum _{v \in V}f(v)$$\end{document}. The minimum weight of a RDF is called the Roman domination number and is denoted by γR(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{R}(G)$$\end{document}. Given a graph G and a positive integer k, the Roman domination problem (RDP) is to check whether G has a RDF of weight at most k. The RDP is known to be NP-complete for bipartite graphs. We strengthen this result by showing that this problem remains NP-complete for two subclasses of bipartite graphs namely, star convex bipartite graphs and comb convex bipartite graphs. We show that γR(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{R}(G)$$\end{document} is linear time solvable for bounded tree-width graphs, chain graphs and threshold graphs, a subclass of split graphs. The minimum Roman domination problem (MRDP) is to find a RDF of minimum weight in the input graph. We show that the MRDP for star convex bipartite graphs and comb convex bipartite graphs cannot be approximated within (1-ϵ)ln|V|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1 - \epsilon ) \ln |V|$$\end{document} for any ϵ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon > 0$$\end{document} unless NP⊆DTIME(|V|O(loglog|V|))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$NP \subseteq DTIME(|V|^{O(\log \log |V|)})$$\end{document} and also propose a 2(1+ln(Δ+1))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2(1+\ln (\varDelta +1))$$\end{document}-approximation algorithm for the MRDP, where Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta $$\end{document} is the maximum degree of G. Finally, we show that the MRDP is APX-complete for graphs with maximum degree 5.


Introduction
Consider G = (V , E) be a simple, undirected and connected graph. For a vertex v ∈ V , the open neighborhood of v in G is N G (v) = {u ∈ V | (u, v) ∈ E} and the closed neighborhood of v is defined as N G [v] = N G (v) ∪ {v}. The degree of a vertex v is |N G (v)| and is denoted by deg (v). The maximum degree of a graph G, denoted by Δ and the minimum degree of a graph, denoted by δ are the maximum and the minimum degree of its vertices. An induced subgraph is a graph formed from a subset D of vertices of G and all of the edges in G connecting pairs of vertices in that subset, denoted by D . A clique is a subset of vertices of G such that every two distinct vertices in the subset are adjacent. An independent set is a set of vertices in which no two vertices are adjacent. A vertex v of G is said to be a pendant vertex if deg(v) = 1 and is called isolated vertex if deg(v) = 0. An edge of G is said to be a pendant edge if one of its vertices is a pendant vertex. A star is a tree on n vertices with one vertex having degree n − 1, called central vertex, and the other n − 1 vertices having degree 1. A comb is a tree obtained by joining a single pendant edge to each vertex of a path. In comb, the path is called backbone and the pendant vertices are called teeth. A bipartite graph G = (X , Y , E) is called tree convex if there exists a tree T = (X , F) such that, for each y in Y , the neighbors of y induce a subtree in T . When T is a star (comb), G is called star (comb) convex bipartite graph [12]. For undefined terminology and notations refer to [21].
A vertex v in G dominates the vertices of its closed neighborhood. A set of vertices S ⊆ V is a dominating set (DS) in G if for every vertex u ∈ V \ S, there exists at least one vertex v ∈ S such that (u, v) ∈ E, i.e., N G [S] = V. A vertex u ∈ V \ S is said to be undominated if N G (u) ∩ S = ∅. The domination number is the minimum cardinality of a dominating set in G and is denoted by γ (G) [8]. The minimum dominating set is a dominating set of minimum cardinality. The MINIMUM DOMINATION problem is to find a dominating set of minimum cardinality.
Roman domination was introduced in 2004 by Cockayne et al. in [3]. A function f : V → {0, 1, 2} is a Roman Dominating Function (RDF) on G if every vertex u ∈ V for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. The weight of a RDF f is the value f (V ) = u∈V f (u). The Roman domination number is the minimum weight of a RDF on G and is denoted by γ R (G). The minimum Roman domination problem (MRDP) is to find a RDF of minimum weight in the input graph.
It is known that RDP is linear time solvable for interval graphs and is NP-complete for planar graphs, bipartite graphs and split graphs [14]. The results obtained in the paper are structured as follows. In Sect. 2, we strengthen the result for bipartite graphs by showing that this problem remains NP-complete for two subclasses of bipartite graphs, i.e., star convex and comb convex bipartite graphs. In Sects. 3, 4 and 5, respectively, we show that RDP is linear time solvable for threshold graphs, chain graphs and bounded tree-width graphs. In Sect. 6, we show that the MRDP for star convex bipartite graphs and comb convex bipartite graphs cannot be approximated within (1 − ) ln |V | for any > 0 unless N P ⊆ DT I M E(|V | O(log log |V |) ). In Sect. 7, we show that the MRDP is APX-complete for graphs with Δ = 5. Finally, in Sect. 8, we give a conclusion.

Complexity results
In this section, we show that the decision version of the Roman domination problem is NP-complete for star convex bipartite graphs and comb convex bipartite graphs by giving a polynomial time reduction from a well-known NP-complete problem, Exact-3-Cover (X 3C) [7], which is defined as follows. EXACT-3-COVER (X3C) INSTANCE A finite set X with | X | = 3q and a collection C of 3-element subsets of X . QUESTION Is there a subcollection C of C such that every element of X appears in exactly one member of C ? The decision version of Roman domination problem is defined as follows. ROMAN DOMINATION PROBLEM (RDP) INSTANCE A simple, undirected graph G = (V , E) and a positive integer k ≤ |V |. QUESTION Does G have a RDF of weight at most k?

Theorem 1 RDP is NP-complete for star convex bipartite graphs.
Proof Given a graph G and a function f , whether f is a RDF of size at most k can be checked in polynomial time. Hence RDP is a member of NP. Now we show that RDP is NP-hard by transforming an instance X , C of X3C, where X = {x 1 , x 2 , . .., x 3q } and C = {c 1 , c 2 , . . ., c t }, to an instance G, k of RDP as follows.
Create vertices x i for each x i ∈ X , c i for each c i ∈ C and also create vertices a, a 1 , a 2 and a 3 . Add edges (a i , a) for each a i and (c i , a) for each c i . Also add edges The graph constructed is shown in the Therefore G is a star convex bipartite graph and can be constructed from the given instance X , C of X3C in polynomial time. Next we need to prove the following claim.
Claim X 3C has a solution if and only if G has a RDF with weight at most 2q + 2.
Proof Suppose C is a solution for X 3C with |C | = q. We define a function f : V → {0, 1, 2} as follows.
Clearly, f is a RDF and f (V ) = 2q + 2. Let k = 2q + 2. Conversely, suppose that G has a Roman dominating function g with weight at most k. Clearly, g(a) + g(a 1 ) + g(a 2 ) + g(a 3 ) ≥ 2. Without loss of generality, let g(a) = 2 and g(a 1 ) = g(a 2 ) = g(a 3 ) = 0. Since (a, c j ) ∈ E, it follows that each vertex c j may be assigned the value 0. We have the following claim.
, which is greater than k, a contradiction. Therefore for each Since each c i has exactly three neighbors in X , clearly, there exist q number of c i 's with

Theorem 2 RDP is NP-complete for comb convex bipartite graphs.
Proof Clearly, RDP is a member of NP. We transform an instance X , C of X 3C, where X = {x 1 , x 2 , . . ., x 3q } and C = {c 1 , c 2 , . . ., c t }, to an instance G, k of RDP as follows.
Create vertices x i , x i for each x i ∈ X , c i for each c i ∈ C and also create vertices a, a , a 1 , a 2 and a 3 . Add edges (a i , a) for each a i and (c j , x i ) if x i ∈ c j . Next add edges (c j , a) and (c j , a ) for each c j . Also add edges by joining each c j to every x i . Therefore G is a comb convex bipartite graph and can be constructed from the given instance X , C of X3C in polynomial time. Next we need to prove the following claim.
Claim X 3C has a solution if and only if G has a RDF with weight at most 2q + 2.
The forward proof is same as in first claim of Theorem 1. Conversely, suppose that G has a RDF g with weight k. This proof is obtained with similar arguments as in the converse proof of Theorem 1 and by using the assignment Now, the following result is immediate from Theorems 1 and 2 .
Theorem 3 RDP is NP-complete for tree convex bipartite graphs.

Threshold graphs
In this section, we determine the Roman domination number of threshold graph.
Although several characterizations are defined for threshold graphs, we use the following characterization of threshold graphs given in [15] to prove that Roman domination number can be computed in linear time for threshold graphs.
A graph G is a threshold graph if and only if it is a split graph and, for split partition (C, I ) of V where C is a clique and I is an independent set, there is an , and there is an ordering {y 1 , y 2 , . . . , y m } of the vertices of I such that

Theorem 4 Let G be a threshold graph. Then
where k is the number of connected components in G.
Clearly, f is a RDF and γ R (G) ≤ k + 1. From the definition of RDF, it follows that Now, the following result is immediate from Theorem 4.

Theorem 5 RDP can be solvable in linear time for threshold graphs.
Proof Since the ordering of the vertices of the clique and the number of connected components in a threshold graph can be determined in linear time [13,15], the result follows.
neighborhoods of the vertices of X form a chain, that is, the vertices of X can be linearly ordered, say is a chain graph, then the neighborhoods of the vertices of Y also form a chain. An ordering α = (x 1 , x 2 , . . . , x p , y 1 , y 2 , . . . , Every chain graph admits a chain ordering [22]. The following proposition is stated in [3].
If G (X , Y , E) is a complete bipartite graph then γ R (G) is obtained directly from Proposition 1. Otherwise, the following theorem holds.
If the chain graph G is disconnected with k connected components G 1 , G 2 , . . . , G k then it is easy to verify that γ R (G) = k i=1 γ R (G i ). Now, the following result is immediate from Theorem 6.

Theorem 7 RDP can be solvable in linear time for chain graphs.
Proof Since the chain ordering and the connected components can be computed in linear time [13,20], the result follows.

Bounded tree-width graphs
Let G be a graph, T be a tree and v be a family of vertex sets V t ⊆ V (G) indexed by the vertices t of T . The pair (T , v ) is called a tree-decomposition of G if it satisfies the following three conditions: (i) V (G) = t∈V (T ) V t , (ii) for every edge e ∈ E(G) there exists a t ∈ V (T ) such that both ends of e lie in V t , (iii) V t 1 ∩ V t 3 ⊆ V t 2 whenever t 1 , t 2 , t 3 ∈ V (T ) and t 2 is on the path in T from t 1 to t 3 . The width of (T , v ) is the number max{|V t | − 1 : t ∈ T }, and the tree-width tw(G) of G is the minimum width of any tree-decomposition of G. By Courcelle's Thoerem, it is well known that every graph problem that can be described by counting monadic second-order logic (CMSOL) can be solved in linear-time in graphs of bounded tree-width, given a tree decomposition as input [2]. We show that RDP can be expressed in CMSOL.

Theorem 9 Given a graph G and a positive integer k, RDP can be expressed in CMSOL.
The CMSOL formula for the RDP is expressed as follows. j( p, q))), where ad j( p, q) is the binary adjacency relation which holds if and only if, p, q are two adjacent vertices of G. Now, the following result is immediate from Theorems 8 and 9 .

Lower bound on the approximation ratio of MRDP in star convex and comb convex bipartite graphs
In Sect. 2, it has been shown that the RDP is NP-complete for the star convex and the comb convex bipartite graphs. In this section, we prove an approximation hardness result for the MRDP in star convex and comb convex bipartite graphs. To show the hardness result for the MRDP, we provide an approximation preserving reduction from the MIN SET COVER problem which is stated below.
Min set cover problem Let X be any non-empty set and C be a family of subsets of X . For the set system (X , C), a set C ⊆ C is called a cover of X , if every element of X belongs to at least one element of C . The MIN SET COVER problem is to find a minimum cardinality cover of X for a given set system (X , C). The following result is proved in [6].
Theorem 11 [6] The MIN SET COVER problem for the input instance (X , C) does not admit a (1 − ) ln |X |-approximation algorithm for any > 0 unless N P ⊆

DT I M E(|X | O(log log |X |) ). Furthermore, this inapproximability result holds for the case when the size of the input collection C is no more than the size of the set X .
Now we are ready to prove the following result:

Theorem 12
The MRDP for a star convex bipartite graph G with n vertices does not admit a (1 − ) ln n-approximation algorithm for any > 0 unless N P ⊆ DT I M E(n O (log log n) ).
Proof In order to prove the theorem, we propose the following approximation preserving reduction. Let X = {x 1 , x 2 , . . . , x p } and C = {c 1 , c 2 , . . . , c q } be an instance of the MIN SET COVER problem. From this, with similar arguments as in Theorem 1, we construct an instance G = (V , E) of MRDP for star convex bipartite graphs. Next, we state the following claim.

Claim MIN SET COVER instance (X , C) has a cover of cardinality m if and only if
G has a RDF of size 2m + 2.
Proof The proof is obtained with similar arguments as in first claim of Theorem 1.
If f is a minimum RDF of G and C * is a minimum set cover of X for the set system (X , C), then f (V ) = 2|C * | + 2. Suppose that the MRDP can be approximated within a ratio of α, where α = (1− ) ln n for some fixed > 0, by using some approximation algorithm, say Algorithm P, that runs in polynomial time. Let k be a fixed positive integer. Then the algorithm SET-COVER-APPROX constructs solution for MIN SET COVER problem. Our algorithm is given in Algorithm 1.

Algorithm 1 SET-COVER-APPROX(X , C)
Require: A set X and a collection C of subsets of X . Ensure: A cover of X . 1: if there exists a cover C of X of cardinality ≤ k then 2: C x = C ; 3: else 4: Construct the graph G; 5: Compute a RDF g on G by using algorithm P; 6: Construct a cover C of X from RDF g (as illustrated in the proof of the Claim in Theorem 12); 7: Clearly, SET-COVER-APPROX runs in polynomial time. If the cardinality of a minimum cover of X is at most k, then it can be computed in polynomial time. Next, we analyze the case, where the cardinality of a minimum cover of X is greater than k. Let C * denotes a minimum cover of X and f be a minimum RDF of G. So, |C * | > k. If C x is a cover of X computed by the algorithm SET-COVER-APPROX, then, |C x | < g(V ) ≤ α( f (V )) ≤ α(2 + 2|C * |) ≤ α(2 + 2 |C * | )|C * |. Therefore, SET-COVER-APPROX approximates a cover of X within a ratio of α(2 + 2 |C * | ).
This proves that the algorithm APPROX-SET-COVER approximates set cover of X within ratio (1 − ) ln p for some fixed > 0. By Theorem 11, if the MIN SET COVER problem can be approximated within a ratio of (1 − ) ln p, then N P ⊆ DT I M E ( p O(log log p) ). It follows that, if MRDP can be approximated within a ratio of (1 − ) ln n for any > 0, then N P ⊆ DT I M E(n O(log log n) ).
Hence, for a star convex bipartite graph G = (V , E), the MRDP cannot be approximated within a ratio of (1 − ) ln n for any > 0 unless N P ⊆ DT I M E(n O(log log n ).

Theorem 13
The MRDP for a comb convex bipartite graph G with n vertices does not admit a (1 − ) ln n-approximation algorithm for any > 0 unless N P ⊆ DT I M E(n O(log log n) ).
Proof The proof is obtained with similar arguments as in Theorem 12, in which replacing the Theorem 1 by Theorem 2 and the first Claim in Theorem 1 by Claim in Theorem 2.

Approximation results
In this section, we obtain an upper on the approximation ratio of the MRDP. We also show that the MRDP is in APX-complete for graphs with maximum degree 5.

Approximation algorithm
In this subsection, we design an approximation algorithm for MRDP based on the well known optimization problem called MINIMUM DOMINATION problem. The following theorem has been proved in [13].
Theorem 14 [13] The MINIMUM DOMINATION problem in a graph with maximum degree Δ can be approximated with an approximation ratio of 1 + ln(Δ + 1).
Let APPROX-DOM-SET be an approximation algorithm that gives a dominating set D of a graph G such that |D| ≤ (1 + ln(Δ + 1))γ (G), where Δ is the maximum degree of a graph G.
Next, we propose an algorithm APPROX-RDF to compute an approximate solution of MRDP. In our algorithm, first we compute a dominating set D of the input graph G using the approximation algorithm APPROX-DOM-SET. Next, we construct a triple T r in which every vertex in D will be assigned with weight 2 and the remaining vertices will be assigned with weight 0. Now, let T r = (D , ∅, D) be the triple obtained by using the APPROX-RDF algorithm. It can be easily seen that every vertex v ∈ V is assigned with weight either 0 or 2. Since D is a dominating set of G, every vertex v ∈ D having weight 0 is adjacent to a vertex u ∈ D having weight 2. Thus, T r gives a Roman dominating function of G. We note that the algorithm APPROX-RDF computes a Roman dominating triple T r of a given graph G in polynomial time. Hence, we have the following result.

Theorem 15
The MRDP in a graph with maximum degree Δ can be approximated with an approximation ratio of 2(1 + ln(Δ + 1)).
Since the MRDP in a graph with maximum degree Δ admits an approximation algorithm that achieves the approximation ratio of 2(1 + ln(Δ + 1)), we immediately have the following corollary of Theorem 15.

Corollary 1
The MRDP is in the class of APX when the maximum degree Δ is fixed.

APX-completeness
In this subsection, we prove that the MRDP is APX-complete for graphs with maximum degree 5. This can be proved using an L-reduction, which is defined as follows.
Definition 1 (L-reduction) ( [16]) Given two NP optimization problems F and G and a polynomial time transformation f from instances of F to instances of G, one can say that f is an L-reduction if there exists positive constants α and β such that for every instance x of F Here, opt F (x) represents the size of an optimal solution for an instance x of an NP optimization problem F. An optimization problem π is APX-complete if: 1. π ∈ APX, and 2. π ∈ APX-hard, i.e., there exists an L-reduction from some APX-complete problem to π .
By using Corollary 1, we can say that MRDP is in APX for graphs with maximum degree 5. To show APX-hardness of MRDP, we give an L-reduction from MINIMUM DOMINATING SET problem in graphs with maximum degree 3 (DOM-3) which has been proved as APX-complete [1].

Theorem 16
The MRDP is APX-complete for graphs with maximum degree 5.
Proof It is known that MRDP is in APX. Given an instance G = (V , E) of DOM-3, where V = {v 1 , v 2 , . . . , v n }, we construct an instance G = (V , E ) of MRDP as follows.
Create n copies of P 3 with b i as the central vertex and a i , c i as terminal vertices. Add the edges {(v i , a i ), (v i , c i ) : 1 ≤ i ≤ n}. Example construction of G from G is shown in Fig. 5.
Note that G is a graph with maximum degree 5. First we need to prove the following claim.
Proof Let G = (V , E), where V = {v 1 , v 2 , . . . , v n } be a graph and G = (V , E ) is a graph constructed from G.
Let D * be a minimum dominating set of G and f : V → {0, 1, 2} be a function on graph G , which is defined as below.

Conclusion
In this paper, we have shown that the RDP is NP-complete for star convex bipartite graphs and comb convex bipartite graphs. Investigating the algorithmic complexity of these problems for other subclasses of bipartite graphs remains open. Next, we have shown that determining γ R (G) is linear time solvable for threshold graphs, chain graphs and bounded tree-width graphs. From approximation point of view, we have given polynomial time approximation algorithm for MRDP and shown that MRDP is APX-hard for graphs with maximum degree 5. The complexity status of this problem is still open for graphs with maximum degree 3 or 4.