## 1 Introduction

The concept of domination has a significant role in graph theory. It has many practical importance in several areas of computer science such as networking, facility location problem, wireless sensor networking, social networking, etc. Let be an undirected graph. A set is said to be a dominating set, if for each vertex , there exist at least one vertex , such that the edge . The dominating set with minimum cardinality is known to be the minimum dominating set, and the corresponding cardinality is known to be the domination number, denoted by of the graph . The domination problem studied intensively in the literature [5, 6, 7]. Recently, researchers started exploring variations on domination to meet the requirement and demand of domination with some additional constraints in several other fields. Some of the variations on domination are Roman domination, Italian domination, perfect Roman domination, perfect Italian domination [1, 9, 8] etc.

This paper mainly focuses on Roman domination. It was first introduced by Cockayne et al. [3] and was motivated from an article, which was based on legion deployment for better security with limited resources [15]. A Roman dominating function (RDF) on graph is defined as a function satisfying the condition that every vertex with is adjacent to at least one vertex with . The weight of a RDF is the value . The Roman domination number (RDN) of a graph , denoted by , is the minimum weight among all possible RDFs on . In the Roman domination problem, for a given undirected simple graph , the objective is to find a Roman domination function such that is minimized. Roman domination problem (RDP) is NP-complete for general graphs [4]. It is also NP-complete when restricted to bipartite graphs, split graphs, and planar graphs [3]. Let be a bipartite graph. The graph is said to be a tree-convex bipartite graph if there exist a tree such that for each , the neighborhood of induces a subtree of . See [14] for linear time algorithm to recognize tree-convex bipartite graph and corresponding tree construction. A graph is a star (comb) convex-bipartite graph if it is a tree-convex bipartite graph and the corresponding tree is a star (comb). A graph is a line convex-bipartite graph if it is a tree-convex-bipartite graph and the corresponding tree is a line graph. In some papers, line convex-bipartite graph is known as convex-bipartite graph. The NP-completeness of star convex bipartite graphs and comb convex bipartite graphs, can be found in [11]. In [11], authors also gave linear time algorithms for bounded treewidth graphs, chain graphs, and threshold graphs. The RDP is linear-time solvable for interval graphs and co-graphs [10]. In [10], authors also gave polynomial-time algorithms for D-octopus graphs and AT-free graphs. The RDP is also studied on circulant graphs, generalized Peterson graphs, and Cartesian product graphs [16]. In the literature, we observed that some problems are NP-complete for bipartite graphs, but the same problems are solvable in polynomial time in some subclasses of bipartite graphs. So, it will be interesting to see the behavior of the Roman domination problem (RDP) in different subclasses of bipartite graphs as mentioned in [12].

The remaining part of this paper is organized as follows. In Section , we introduce some relevant preliminaries along with some of the important observations and lemmas related to line convex bipartite graph and Roman domination. In Section , we detail the approach for finding the Roman domination function of a line convex bipartite graph and the corresponding algorithm. Finally, we conclude the paper in section .

## 2 Preliminaries

Let be a bipartite graph, where and are ordered from top to bottom. is said to be a line convex bipartite graph if it is a tree convex bipartite graph and the corresponding tree is a line graph. In other words, a bipartite graph is said to be a line convex over the vertices of partite set if there exist a linear ordering of the vertices of such that for each vertex , neighbors of form an interval in i.e., , where is the index associated with the lowest indexed neighbor of , is the index associated with the highest indexed neighbor of and is the interval associated with the vertex . The bipartite graph in Fig. 1(a) is a line convex bipartite graph as the vertices of the partite set can be rearranged in such a way that the is an interval, for each as shown in Fig. 1(b). In this paper, whenever, we refer to line convex bipartite graph that means the graph is convex with respect to . The line convex bipartite graph is interchangeably referred to as convex bipartite graph.

###### Lemma 2.1.

###### Observation 2.2.

Given a graph , each isolated vertex of the graph carries Roman value in any optimal solution of the Roman dominating function.

###### Observation 2.3.

Let be a line convex bipartite graph. If is a bipartite graph obtained from by adding two isolated vertices and , then is also a line convex bipartite graph and .

###### Lemma 2.4.

Given a line convex bipartite graph satisfying Lemma 2.1, any induced graph , where , , , and are two isolated vertices. The vertices that appear in between the neighbors of but are not the neighbors of are isolated.

###### Proof.

The induced graph is a line convex bipartite graph with respect to not necessarily with . So all the neighbors of may not be consecutive. Suppose there exist a vertex, say that appears in between the neighbors of and is not the neighbor of but is not isolated. In that case, must be connected (through an edge) to a vertex , where , i.e., as shown in Fig. 3(a). That means there exist at least one neighbor (say, w) of which lies above . As is connected to with an edge, so the interval must contain (see Fig. 3(b)), i.e., is also connected to with an edge (Lemma 2.1). Therefore, becomes one of the neighbor of , which leads to a contradiction.

∎

###### Lemma 2.5.

Given a line convex bipartite graph satisfying Lemma 2.1, where , . If , then in an optimal solution the Roman value associated with the pair will never be or .

###### Proof.

Suppose in an optimal solution, and , . Now, we can always get a better solution by reassigning to , which is a contradiction. The same can be claimed for , also. ∎

###### Lemma 2.6.

Given a line convex bipartite graph satisfying Lemma 2.1, where , . If , then in an optimal solution the Roman value associated with the pair will never be .

###### Proof.

Suppose in an optimal solution and , , then there must exist at least one vertex in and another vertex in which will Roman dominate and , respectively. If it is so, then there must exist two edges and , and they must cross each other. Due to Lemma 2.1, must have edge with each vertex starting from to , i.e., , which is a contradiction. ∎

###### Lemma 2.7.

Given a line convex bipartite graph satisfying Lemma 2.1, where , . If , then in an optimal solution the Roman value associated with the pair will never be or .

###### Proof.

If , and , then in an optimal solution will never be isolated, otherwise the solution is not optimal. Now, since is not isolated, it is connected (with an edge) to at least one vertex (say, ) in and dominates one/more vertices from the partite set but not (as ). Unlike , should be dominated by at least one vertex (say, ) in otherwise the graph is not Roman dominated. That means the edges and are crossing each other and lies above to (index associated with vertex is less than ). Due to Lemma 2.1, forms an interval including , i.e., , which leads to contradiction. The same can be claimed for also. ∎

###### Lemma 2.8.

Given a line convex bipartite graph satisfying Lemma 2.1, where , . If , then in an optimal solution the Roman value associated with the pair will never be .

###### Proof.

If and , then and will never be isolated. Otherwise the solution will not be optimal. That means must be dominating at least one vertex (say, q) from the partite set but not (as ) and similarly, must be dominating at least one (say, p) from but not , i.e., . Therefore, the edges and should be crossing each other. Due to Lemma 2.1, must form an interval including , i.e., , which leads to a contradiction. ∎

## 3 Algorithm for Roman domination in convex bipartite graphs

In this section, we present a polynomial time algorithm to find an optimal Roman domination function for a given convex bipartite graph .

### 3.1 Notations

Let be a convex bipartite graph. Let and such that (respectively, ) arranged from top to bottom. We add two vertices and into the set and , respectively such that (respectively, ) is above (respectively, ). In this paper, the given graph is assumed to be line convex with respect to . For , let assigns Roman value to the vertex , means has Roman value . In this paper, and are interchangeably used. The lowest index neighbor of is denoted by , and defined by . The highest index neighbor of is denoted by , and defined by , where is the open neighborhood of . Let be the induced subgraph consisting of vertices , , …, in the partite set and , , …, in the partite set . Let be the set of edges in the induced subgraph . Assume is the optimal Roman domination number (RDN) of . Some other important notations related to the induced graph are as follows:

Optimum Roman domination number of when the Roman value associated with the pair is and .

Optimum Roman domination number of when the Roman value associated with the vertex is and .

Optimum Roman domination number of when the Roman value associated with the pair is and .

Optimum Roman domination number of when the Roman value associated with the vertex is and .

Optimum Roman domination number of when the Roman value associated with the pair is and .

Optimum Roman domination number of when the Roman value associated with the pair is and .

Optimum Roman domination number of when the Roman value associated with the vertex is and .

Optimum Roman domination number of when the Roman value associated with the vertex is and .

### 3.2 Overlapping subproblem and optimal substructure

Each induced graph is a subgraph of the given line convex bipartite graph and the Roman domination problem (RDP) on the induced subgraphs can be viewed as a subproblem of RDP on and their minimum RDF can be used to calculate the RDF of . Given a line convex bipartite graph, the RDF corresponding to each induced graph (where and ) can be calculated recursively and stored for further use. As a preliminary step, the vertices of the line convex bipartite graph are reordered based on Lemma 2.1. We added isolated vertices intentionally to the graph ; by doing so, the convexity property is not hampered (Observation 2.3). It helps to meet the base condition.

To begin with, the pair of isolated vertices, and are taken and (where , ) is the first subgraph under consideration. As and are isolated vertices, so and , and hence RDN of is (Observation 2.2). Next onwards, each time a new vertex (consecutive to the previously added vertex) is added and the corresponding induced graph is considered. While finding the RDF of the induced graph , the pair of vertices with highest indices in each partite set, i.e., is considered. In , either (A) or (B) . Now, we consider both cases separately.

Case A ): We consider all possible Roman domination values of and and choose the best solution.

case 1: ,

Since Roman domination values of both and are , (respectively, ) is dominated by a vertex (respectively, ) in the partite set (respectively, ), where (respectively, ) as shown in Fig. 4(a). If such or does not exist, then case is invalid. Here, we consider Roman value of each vertex in the pair , where and equal to to solve the problem. In worst case there will be such pairs, i.e., number of subproblems. In each subproblem Roman value to will dominate all the vertices in and all the vertices in are consecutive, whereas Roman value to will dominate all the vertices in but may not be consecutive (see Fig. 4(a)). So those vertices (if any) that appear in between the neighbors of need to be handled separately (see Algorithm 1) ensuring optimality (see Theorem 3.2). The sub-problems embeded within each pair of will be different from each other due to the variation in edges in their respective open neighborhood. For a specific pair of , the number of sub-problems under consideration depends on their open neighborhood. Hence, the Roman value associated with is as follows:

where , , . (refer Algorithm 1) function finds the optimal Roman cover of the vertices that are not the neighbors of but physically appear in between the neighbors of as the vertices in each partite set in follows a particular order i.e. and .

Let the set of vertices that appears in between the neighbors of but are not the neighbors of be , be the neighbors of except (as ), i.e., and be the corresponding induced graph, where . See Algorithm 1, which finds the optimal Roman domination of with respect to the induced graph .

For each vertex (except the isolated vertices, if any), is an interval in and the intervals are already sorted in non-decreasing order with respect to their last index (Lemma 2.1). Let be the set of all such intervals, i.e., and let fIndex be the index associated with the vertex where an interval ends. Lines in Algorithm 1 find the minimum number of vertices in which stabs all intervals (see Theorem 3.1) and lines assign the optimal Roman value (see Theorem 3.1).

###### Theorem 3.1.

The set in Algorithm 1 finds the minimum number of vertices in , which stabs all intervals.

###### Proof.

On contrary, suppose the vertex is not in the optimum solution. Then the first interval might be stabbed by a vertex that is present either above or below in the optimal solution. It can not lie below to because if it is so, then the first interval is still not stabbed as is the last vertex of the first interval. If it is present above to it, then it can always be replaced by and is an optimal solution. Hence, the vertex must be in at least one optimal solution. ∎

###### Theorem 3.2.

Algorithm 1 gives optimal Roman coverage of with respect to .

###### Proof.

From Observation 2.2, each isolated vertex (if any) in has Roman value (lines ). The remaining non-isolated vertices in can be covered by choosing vertices (i.e., the set ) from optimally (see Theorem 3.1). Lines in Algorithm 1 finds the minimum number of vertices i.e., in that covers all vertices in except the isolated vertices if any. So the optimal Roman value associated with can be calculated by assigning and , where , , and , . ∎

###### Lemma 3.3.

The set of vertices (that are neighbors of or ) with Roman value in subproblem will have Roman value in subject to .

###### Proof.

WLOG, let be a vertex with Roman value in and also a neighbor of . Since , the Roman value of is (Lemma 2.5) in . ∎

The terms and ensure the optimality (see Lemma 3.3) of by resetting the Roman value of all such vertices that are neighbors of or with Roman value in to .

case 2: , for

In this case, dominates itself only as , so the optimal solution of can be directly calculated from . The sub-problem decides whether the Roman value of is , or in the optimal solution if . However, due to Lemma 2.5, will never arise. Hence, the optimal Roman value for can be expressed in terms of as follows:

case 3: ,

Since has Roman value , it dominates all the vertices in as shown in Fig. 4(b) and all vertices in will be consecutive due to convexity of . The neighbors of may have Roman values or but not (see Lemma 2.5). We set Roman value instead of to those vertices in the subproblem with Roman value , where and are the neighbors of . Hence, the term is subtracted from , where . Thus, in this case can be expressed as follows:

where

case 4: , for :

In this case, dominates itself only as . The optimal solution of can be calculated directly from . The solution of the sub-problem sets the Roman value of to either , or in the optimal solution. However, due to Lemma 2.5, will never arise. Hence, can be expressed as:

case 5: ,

Since has Roman value , it covers all the vertices including as shown in Fig. 5(a). Since, may not follow convexity, so the vertices (if any) that appear in between the neighbors but are not the neighbors of are isolated (Lemma 2.4) in the induced graph . The explanation for the term is similar to case , where . Hence, can be expressed in terms of as:

where ,
assigns Roman value to the isolated vertices that fall in between the neighbors of but are not the neighbours of .

case 6: ,

In this case, and will cover all their respective neighbors as shown in Fig. 5(b). Now, there may exist some isolated vertices in between the neighbors of (Lemma 2.4). assigns Roman value to those vertices. The explanation for the subtracted terms and is similar to Case , where . In this case, can be expressed as follows:

where assigns Roman value to the isolated vertices that fall in between the neighbors of but are not the neighbors of .

Case B (): The possible Roman function assignments from the set to and may be one of the followings:

case 1: , for :

This case will never arise due to Lemma 2.6 when and Lemma 2.7 when .

case 2: , for :

This case can be handled similar to Case A2:
.

case 3: , for :

This case can be handled similar to Case A4: .

case 4: , for :

This case will never arise due to Lemma 2.7 when and Lemma 2.8 when .

Hence, by observing all the cases, we have the following recursive equation:

### 3.3 Algorithm

This algorithm MRDN-ConBipGraph (see Algorithm 2) finds the minimum Roman domination number of a line convex bipartite graph , where , , and , with the assumption that the graph satisfies Lemma 2.1.

###### Lemma 3.4.

The time complexity of Algorithm 2 is polynomial.

###### Theorem 3.5.

The Roman domination number obtained from MRDN-ConBipGraph for the graph is an optimal solution.

###### Proof.

We prove the theorem using induction on number of vertices. Given a convex bipartite graph, satisfying Lemma 2.1, where is the set of vertices with , and as the set of edges. The graph obtained by adding two isolated vertices and to each partite set of is also a convex bipartite graph (Observation 2.3), say with , and . While proving the theorem, we have used the lexicographic ordering^{1}^{1}1

Lexicographic ordering: The ordered pair

is less than or equal to if either , or and .[13]. Base case: : and are two isolated vertices, hence their optimal Roman values are (Observation 2.2), i.e., and . Hence, and is optimal. Inductive step: Let be the optimal RDN of , then we can always find the optimal solution for , where in lexicographic ordering and . Here, we are considering all possible Roman values of and , where and and find optimal solution in each of the cases (see Section ). Finally, we are choosing best solution. Hence, the theorem follows. ∎## 4 Conclusion

Here, we have considered Roman domination problem in line convex bipartite graph and proposed a polynomial time algorithm for it. There exist some other subclasses of bipartite graph such as circular convex bipartite graphs, chordal bipartite graphs, triad convex bipartite graphs etc. for which the status of Roman domination is still unknown, and it will be interesting to see whether poly-time algorithms exist for these graphs or not.

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