The motivation for the problem we study stems from the problem of deployment of military forces to guard several points of interest, modeled by an undirected graph. Such problems from different historical eras were described by ReVelle and Rosing  (see also Stewart ). For instance, the authors describe a defense-in-depth strategy by Emperor Constantine (Constantine the Great, 274–337) where units were deployed such that any city without a unit was to be neighbored by a city harbouring two units. The idea was that if the city without a unit was attacked, the neighboring city could dispatch a unit to protect it without becoming vulnerable itself. In this setting, the objective was to minimize the total number of units needed. Albeit overly simplified particularly for the modern era to be of practical use, these type of domination problems on graphs have resulted in interesting graph-theoretical problems that have attracted significant interest from the research community.
Let be a simple undirected graph. To reduce clutter, we can write an element as . The open neighborhood of a vertex , denoted by , is the set of neighbors of excluding itself, i.e., . The degree of a vertex is the number of edges incident to it, i.e., . In particular, a vertex of degree one is a pendant and the vertex adjacent to a pendant vertex is a support. For the following discussion, let be a vertex-labeling of .
We say that is a perfect Italian dominating function on , abbreviated a PID-function, when it holds that whenever for any , it holds that , i.e., the accumulated weight assigned to the neighbors of by is exactly two. The weight of is the sum of its labels, i.e., . The perfect Italian domination number of , denoted by , is the minimum weight of a PID-function on . This concept was introduced by Haynes and Henning  as a natural variant of similar, previously rather heavily-studied, parameters of so-called Roman domination introduced by Cockayne et al. . We refer the interested reader to e.g., [12, Section 3.9] for a brief overview of some of these variants, but describe some relevant to our work in the following.
We say that is a Roman dominating function, abbreviated an RDF-function, on if every vertex for which is adjacent to at least one vertex for which . The Roman domination number of , denoted by , is the minimum weight of an RDF-function on . While introducing the concept, Cockayne et al.  also gave several bounds for and determined its value for certain structured graph classes including paths, cycles and complete multipartite graphs. For example, the authors proved that and that implies to be edgeless, where is the domination number of . Further, they mentioned that is has been proved that deciding whether a graph admits an RDF-function of weight at most is NP-complete. For further combinatorial results on , see the survey [6, Section 5.7]. A possible application in network design is described by Chambers et al. , while Liedloff et al.  give algorithms for several structured graph classes.
Another variant of perfect Italian domination, introduced by Chellali et al. , is obtained by relaxing the constraint so that for every , if , then , i.e., the accumulated weight of assigned to the neighbors of is at least two. Such an is known as a Roman -dominating function of , also referred to as an Italian dominating function by Henning and Klostermeyer . Here, the Roman -domination number of , denoted by , is the minimum weight of a Roman -dominating function on . In addition to various combinatorial results, Chellali et al.  also proved that deciding whether a graph admits a Roman -dominating function of weight at most is NP-complete even when is bipartite.
We continue the study of perfect Italian domination initiated by Haynes and Henning  by giving the following results.
In Section 2, we relate the perfect Italian domination number to other well-known Roman domination numbers. Further, we characterize the graphs such that which includes connected threshold graphs, paths, cycles, and wheels. We proceed to give a characterization of graphs such that , and then conclude by determining the exact value of the parameter for complete multipartite graphs.
In Section 3, we consider the question of Haynes and Henning  for finding best possible upper bounds on as a function of the order when is planar or regular. For planar graphs and split graphs, we prove that there is an infinite family of such connected graphs such that , meaning that no upper bound of the form exists, for any . For cubic graphs, we prove that , and demonstrate that these bounds are tight.
In Section 4, we turn to complexity-theoretic questions. Specifically, we prove that deciding whether a given graph admits a PID-function of weight at most is NP-complete, even when is restricted to the class of bipartite planar graphs. We also strengthen the result of Chellali et al.  by showing that deciding whether admits a Roman -dominating function of weight at most is NP-complete, even when is both bipartite and planar.
We conclude in Section 5 by giving some further open problems and conjectures arising from our work.
2 Basic bounds, properties and characterizations
In this section, we determine some basic properties of the perfect Italian domination number of a graph.
2.1 Graphs with perfect Italian domination number two
We begin with the following known bounds.
Theorem 1 (Chellali et al. ).
For every graph , it holds that .
For every graph , it holds that .
Every PID-function of is a Roman -dominating function of , so the bound follows. ∎
Clearly, the optimal PID-function of a graph consists of optimal PID-functions of its components, as made precise in the following.
If is a disconnected graph with components , then .
For every integer , it holds that and .
The following observation characterizes the graphs with . Recall that the join of graphs and is the graph union of and with all the edges between and added.
A non-trivial connected graph has precisely when can be written as the join of and , where is either , or .
For to have , there must exist a PID-function that labels (i) exactly one vertex 2 and the rest 0 or (ii) exactly two vertices 1 and the rest 0. If exactly one vertex has label 2, all vertices distinct from must be adjacent to it, i.e., must be . Similarly, if there are two vertices and with label 1, must be either or meaning that dominates at least and vice versa for . ∎
Several structured graph classes fall under the above characterization, as we will see next.
A non-trivial connected threshold graph has .
Every threshold graph can be represented as a binary string , read from left to right, where 0 denotes the addition of an isolated vertex and 1 denotes the addition of a dominating vertex (for a proof, see [18, Theorem 1.2.4]). Because is connected, the last symbol of is a 1. As has a dominating vertex, the proof follows by Proposition 5. ∎
The following results are now immediate, where , , and denote the star graph, complete graph, and wheel graph, respectively, on vertices.
For every integer , it holds that .
For every integer , it holds that .
For every integer , it holds that .
For every integer , it holds that .
The graph can be written as the join of and (i.e., the edgeless -vertex graph), so the proof follows by Proposition 5. ∎
2.2 Bounds via fair domination
In this subsection, we give a characterization of graphs with . In order to do so, let us first introduce some concepts from domination.
Let be a graph. For , a -fair dominating set of is a dominating set such that for every . That is, every vertex not in has precisely neighbors in . The -fair domination number of , denoted by , is the minimum cardinality of a -fair dominating set in . This concept was introduced by Caro et al.  (see also ). It is also captured by the concept of -domination as introduced by Chellali et al. . Here, a subset is a -set if for every vertex it holds that , that is, every vertex not in has at least but no more than neighbors in . Clearly, a -fair dominating set is equivalent to a -dominating set. Finally, such a set is also known as a perfect -dominating set (see e.g., [3, 4]).
For every graph , it holds that .
Let be a 2-fair dominating set. Construct a vertex-labeling such that for and for . By definition, every for which it holds that there are precisely two vertices with in , so is a PID-function. The weight of is which can be as small as , completing the proof. ∎
In order to exploit the previous theorem, we prove the following result regarding the structure of any PID-function witnessing .
Any PID-function of a graph witnessing uses exactly three ones and no twos.
Suppose this was not the case, i.e., that instead set and for some distinct . Now consider any such that . Because is a PID-function of weight three, it must hold that . But because is adjacent to and , the labels on the neighbors of assigned by cannot sum to exactly two, contradicting the fact that is a PID-function. ∎
We are now ready to prove the main result of the section.
A graph with has if and only if has a 2-fair dominating set of size 3.
Suppose that . By Lemma 12, any PID-function of has picked three vertices, say , , and such that and labeled every other vertex 0. We claim that is a 2-fair dominating set of size 3. Indeed, every vertex with label 0 must be adjacent to exactly two vertices of since is a PID-function, so the claim follows.
For the other direction, construct a PID-function from a 2-fair dominating set such that for and for . Clearly, as is a 2-fair dominating set, every is adjacent to exactly two vertices labeled 1. Further, because , we have that . As , we conclude that . ∎
It is also possible to state the same result in a different way. To do this, we observe the following.
Let be a connected graph. A subset of size is an -fair dominating set in if and only if is an -fair dominating set in .
A 1-fair dominating set is also known as a perfect dominating set (see Fellows and Hoover ).
Let be a connected graph. A subset of size three is a 2-fair dominating set in if and only if is a perfect dominating set in .
We can then restate our earlier theorem as follows.
A graph with has if and only if has a perfect dominating set of size 3.
Let us then proceed to determine the perfect Italian domination number of complete multipartite graphs.
For every two integers , it holds that .
Let us denote . As does not have a pair of vertices that dominate every vertex (possibly excluding each other), it follows by Proposition 5 that . The complement of is a disjoint union of two cliques and . Thus, does not admit a perfect dominating set of size three, so . A matching upper bound is given by an which assigns and for one and one , while setting remaining labels to 0. This completes the proof. ∎
For every three integers , it holds that .
For integers , it holds that .
Let us denote .
For the sake of contradiction, assume that a PID-function of with weight less than exists.
In other words, there must exist a vertex in a set of the -partition of for some with .
Let us consider all the possibilities as to how the neighbors of must be labeled.
Case 1: There is a neighbor of in with such that .
Because is a PID-function, it follows immediately that every vertex in must be labeled 0.
Furthermore, every vertex with label 0 in must have neighbors of weight exactly two.
Consequently, each vertex of has label 2.
But now a vertex of label 0 in for any has neighbors of weight four contradicting the fact that is a PID-function.
Case 2: There are neighbors and of in with such that .
Because , there is a vertex distinct from and in whose neighbors in must have weight two.
But similarly to Case 1, there is then a vertex in with label 0 whose neighbors have weight four, a contradiction.
Case 3: There are neighbors and of in and , respectively, with such that .
Similarly to Case 1, we again observe that every vertex in must have label 0. Now, for instance, a vertex in with label 0 requires that a vertex in distinct from has label 1. But then a vertex of distinct from has weight three, a contradiction. ∎
The previous lemmas together prove the following.
Let be the complete -partite graph, where for each . Then
The complete multipartite graph for shows that the difference between and can be made arbitrarily large. Indeed, by Theorem 20 we have that , but as witnessed by labeling exactly one vertex 1 from three different sets of the -partition of and labeling the remaining vertices 0.
3 On upper bounds for restricted graph classes
Haynes and Henning  proposed the problem of determining the best possible constant such that for all -vertex graphs belonging to a particular class of graphs. In particular, they showed that if is the class of connected bipartite graphs, then , whereas if is the class of trees (on at least 3 vertices), then . Further, the authors suggested to study the problem further when would be e.g., the class of planar graphs or regular graphs.
In the following subsections, we settle precisely the question when is the class of connected planar graphs by proving, perhaps surprisingly, that . In addition, we also completely settle the question when is the class of connected cubic graphs by proving that . Further, when is the class of -regular graphs for , we show that . When , this family is also connected. We conclude by observing that when is the class of connected split graphs, implying that also when is any superclass of split graphs, like the class of chordal graphs.
3.1 Planar graphs
In this subsection, we describe an infinite family of connected planar graphs that have , thus proving that when is the class of connected planar graphs.
Let be the connected 10-vertex planar graph that is formed by adding two dominating vertices to and then finishing by connecting a pendant vertex to every vertex except for two vertices of degree three (see Figure 1). In particular, name the four support vertices of so that and are those with degree five, and and are those with degree four. The graph is obtained via widening by connecting both and with the pendants of and , say and , respectively, and by introducing a new pendant vertex to both and . The widening of to obtain is illustrated in Figure 1. In total, a widening operation adds two vertices and six edges. In general, the graph for any is obtained recursively by widening , which in turn is obtained by widening , and so on. Our goal is to show that . To this end, we make the following claims concerning any PID-function with weight less than .
Let be a PID-function of with weight less than . It must hold for the support vertices and that .
If this was not the case, i.e., if , none of the unlabeled non-pendant vertices could be labeled 0 because and are in the neighborhood of each such vertex. Thus, the weight of any PID-function would then be at least . Further, must label every remaining unlabeled vertex 0. This means that every support vertex must be labeled 2 (for otherwise is not a PID-function), but then the weight is . ∎
Let be a PID-function of with weight less than . The function must label and .
For the sake of contradiction, suppose that has weight less than and . Because is a PID-function, it holds that . Clearly, the pendant of cannot be labeled 0, so first suppose that pendant of was labeled 2. It follows that every other vertex adjacent to must be labeled 0. But now it must be the case that and , but , contradicting the fact that is a PID-function. So it must be the case that the pendant of is labeled 1. It follows that precisely one unlabeled neighbor of is labeled 1 while the rest are labeled 0. Now, observe that there exists a non-neighbor of whose all neighbors have been labeled 0 except for . Thus, it must be that . But now a neighbor of , labeled 0, is adjacent to (with label 1) and (with label 2), contradicting the fact that is a PID-function. We conclude that . By a symmetric argument, under any valid PID-function whose weight is less than . ∎
For any integer , it holds that .
For the sake of contradiction, suppose that there is a PID-function for for any with weight less than . By combining Lemma 23 with Lemma 24, we know that any such must label . Consider any vertex that is a common neighbor of both and . If , all neighbors of must also be labeled 0. In particular, it now holds that but then the pendant vertices and cannot receive any of the labels 0, 1, or 2 without violating the fact that is a valid PID-function, a contradiction. Otherwise, if there is no such with , the weight of is at least with only the pendants unlabeled. Clearly, the two pendants of and cannot be labeled 0, but can be labeled 1. For the pendants and of and there are two choices: either set (i) and or set (ii) , and similarly the same for and . In both cases has weight , a contradiction. ∎
The previous lemma establishes the main result of this subsection.
There is an infinite family of connected planar graphs such that .
As a side remark, we can also see that for any , the treewidth of is three. Thus, unlike for e.g., chromatic number, it is not true that the perfect Italian domination number of a graph could be bounded as a function of treewidth.
3.2 Regular graphs
In this subsection, we shift our focus to regular graphs. As a main result here, we derive tight upper and lower bounds for the perfect Italian domination number of cubic graphs.
A strong matching, also known as an induced matching, is a set of edges of a graph such that no two edges in are connected by an edge of . Viewed differently, an induced matching is an independent set in the square of the line graph . The strong matching number, denoted by , is the size of a maximum induced matching of . For the next lemma, the key observation is that if is a strong matching in a cubic graph , then is a 2-fair dominating set of .
Every cubic graph has .
Let be any strong matching of . Construct a vertex-labeling such that for every and label all other vertices 1. Clearly, is a PID-function since every vertex with has two neighbors labeled 1 and one labeled 0. The weight of is , which is equal to when . ∎
The following bound for the strong matching number will be useful for us.
Theorem 28 (Joos et al. ).
A cubic graph with edges has .
A connected graph on vertices with maximum degree has .
We are now ready to establish the main result of this subsection.
Every connected cubic graph with vertices has . Moreover, these bounds are tight.
The lower bound follows from Theorem 29 by having . The claimed upper bound follows by applying Lemma 27 for which we combine the fact that every cubic graph with vertices has edges with Theorem 28. That is, we see that
To see that the lower bound is tight, one can consider any connected cubic graph with 8 vertices. For instance, when is the 8-vertex cubical graph, we have that . To see that the upper bound is tight, one can consider defined as the Cartesian product of and . Clearly, does not satisfy the condition of Proposition 5. Further, is isomorphic to the 6-cycle, which does not admit a perfect dominating set of size three, so by Theorem 16 it holds that . By our upper bound as well, so both bounds are tight. ∎
Another example to see that is tight is the 6-vertex cubic graph obtained by taking a and making a new vertex adjacent to each of the three vertices in the other set of the bipartition.
For every there is an infinite family of -regular graphs such that . For every , this family is connected.
For the claim is clear as indicated by Theorem 20.
Now, for we turn to a computer search with the help of House of Graphs , an online database for “interesting” graphs. Here, if we can find a -regular graph for which , an infinite (disconnected) family such graphs is obtained by taking multiple disjoint copies of . Indeed, by Proposition 3, an optimal PID-function for such a graph will also have weight . Below is a list of graphs represented in the well-known graph6 code:
The first of three is 6-regular, the second is 7-regular, and the third 8-regular, all with the property that they do not admit a PID-function of weight less than . ∎
While for every , there are -regular graphs with , we conclude with the following observations.
Let be a -regular graph for any and let be a PID-function of . Every such that is adjacent to or vertices such that .
If this was not the case, the sum would be not equal to two contradicting the fact that is a PID-function. ∎
For every , there does not exist an -vertex -regular graph with .
To reach a contradiction, let be an optimal PID-function of witnessing that . If such a graph existed, then a vertex with would be adjacent to exactly vertices with , contradicting Lemma 32. ∎
3.3 Split graphs
In this subsection, we consider split graphs defined as graphs whose vertex set can be partitioned into a clique and an independent set. Split graphs are highly restricted graphs forming a subclass of chordal graphs, which in turn are a subclass of perfect graphs.
For any , let be the split graph obtained by starting from and by choosing four distinct arbitrary vertices of it and adding two new vertices and with the edges (see Figure 2). That is, forms an independent set, while induces a clique of size .
For any , it holds that .
For the sake of contradiction, suppose that and that this is witnessed by a PID-function . Because has weight less than , there must exist at least one vertex such that . Suppose that . Then, without loss of generality, there are two possibilities: either (i) and or (ii) and . In both cases, it follows that all the other vertices of the must be labeled 0 by . In particular, it holds that , but now there is no label can assign to . Thus, .
Without loss of generality, suppose that . Now, if , it must be that . Again, by the same argument as above, there is no label can assign to . Thus, if then must hold. Now, must label exactly one vertex of the vertices of the with 1 and the other with 0. But then there is always at least one vertex in , which is distinct from as , such that , contradicting the fact that is a PID-function.
Because none of , , and can be labeled 0 by , it follows that , and thus for every . At this point, the only possibility is that . It follows that . As no other vertex can be labeled 0, we can label every remaining vertex 1. But now the weight of is , a contradiction. We conclude that , which is what we wanted to prove. ∎
The previous lemma establishes the following result.
There is an infinite family of connected split graphs such that .
We can further contrast this result with the fact that threshold graphs, which are precisely the -free split graphs, always admit a PID-function of weight at most 2 by Proposition 6.
4 Hardness of perfect Italian domination
In this section, we prove that perfect Italian domination is NP-complete, even when restricted to bipartite planar graphs. In all our hardness proofs, we omit explicitly showing membership to NP as it is an easy exercise.
To prove the claimed result, we give a polynomial-time reduction from Planar Exact Cover by 3-Sets in which we are given a finite set with and a family of 3-element subsets of . The goal is to decide whether there is a subfamily of such that every element of appears in exactly one element of . Every instance is associated with a bipartite incidence graph, in which the first set of the bipartition corresponds to elements in and the second to elements in . The edge set is defined such that two vertices are connected precisely when an element of is contained in an element of . In Planar Exact Cover by 3-Sets, we have the further constraint the incidence graph is both bipartite and planar. This problem was shown to be NP-complete by Dyer and Frieze .
Theorem 36 (Dyer and Frieze ).
Planar Exact Cover by 3-Sets is NP-complete.
Before describing our reduction, let us introduce the following gadget. For any positive integer , the fish gadget is constructed by starting from the disjoint union of vertices partitioned into two equally-sized sets and , and by adding two vertices and such that is adjacent to every vertex in and is adjacent to every vertex in . Thus, has a total of vertices, with vertices of degree two and vertices of degree one. The fish gadget is illustrated in Figure 3.
For any , any PID-function of has weight at least if . Similarly, if , has weight at least .
We say that a vertex for which is satisfied if . Even more precisely, we say that such a is out-satisfied (with respect to some subgraph of ) if . Similarly, is in-satisfied if . For the following statement, the subgraph is to be understood to be the gadget itself.
For any , any PID-function of that sets has optimal weight 2 if is out-satisfied. Otherwise, if is in-satisfied, has optimal weight 4.
In the first case, set and label other vertices 0. In the second case, set , label an arbitrary vertex in with 2, and label other vertices 0. ∎
Let us call Perfect Italian Domination the problem where we are given a graph and an integer , and the goal is to decide whether admits a PID-function of weight at most .
Perfect Italian Domination is NP-complete for bipartite planar graphs.
Let be an instance of Planar Exact Cover by 3-Sets, such that , , and . We proceed by describing a polynomial-time reduction to Perfect Italian Domination as follows.
Let be the bipartite incidence graph of , which we can also safely assume to be planar by Theorem 36. So more precisely, with and adjacent precisely when is a member of . Let . To obtain from , identify for with a fish gadget (at its vertex ) and attach to for two pendants and . We tacitly name the vertex of a fish gadget corresponding to the vertex . Clearly, because the fish gadget is both bipartite and planar, is bipartite and planar as well. We claim that has an exact cover if and only if admits a PID-function of weight at most .
Let be an exact cover of . We construct a vertex-labeling of such that for ; all other vertices not in are labeled 0. Here, if , we set and if , then . At this point, the labels used by have weight . For each , we label , , and all the remaining vertices 0. As and , the weight of is exactly . Now, since is an exact cover, every is out-satisfied by some corresponding to a . For each , if , then is satisfied by . It follows that is a PID-function.
Conversely, suppose that is a PID-function of weight . It holds for every that for otherwise would have weight at least by Proposition 37. Further, as requires labels of weight at least , it follows by Proposition 38 that each must be out-satisfied for otherwise would have weight at least . It follows that has allocated labels of weight to . Further, this is only possible if for for otherwise would have weight at least . Therefore, since is a PID-function, every is out-satisfied by exactly one for which . Consequently, is an exact cover of . ∎
It is worth mentioning that the earlier result of Chellali et al. [8, Theorem 18] regarding the hardness of computing also works for bipartite planar graphs. Let us call Roman -Domination the problem of deciding whether given a graph and an integer , it is true that .
Roman -Domination is NP-complete for bipartite planar graphs.
Chellali et al. [8, Theorem 18] prove NP-completeness of Roman -Domination for bipartite graphs by a polynomial-time reduction from an arbitrary instance of Exact Cover by 3-Sets. In short, their reduction begins from the bipartite incidence graph of , but replaces every vertex corresponding to a with a with a chord followed by a 2-vertex path. Because this gadget is both bipartite and planar, we ensure that the instance of Roman -Domination is both bipartite and planar by assuming that is planar. By Theorem 36, we can do this safely, so the result follows. ∎
5 Open problems
In this section, we conclude by highlighting some open problems arising from our work.
We begin with the following complexity-theoretic statement.
For every , Perfect Italian Domination is NP-complete for the class of -regular graphs.
In the light of our construction in the proof of Theorem 26, it might be interesting to consider other planar graphs with . We verified by a computer search the smallest planar graph with to have vertices, and there are no other such planar graphs on 7 vertices. Thus, one might ask the following.
Can we characterize the connected planar graphs such that , or at least find some conditions for this to hold?
Also, after Theorem 26, it is natural to raise the question of Haynes and Henning  for the class of bipartite planar graphs. At the same time, given our NP-completeness result Theorem 39, one should not expect a polynomial-time characterization for this class.
Determine the best possible constant such that for all -vertex graphs belonging to the class of connected bipartite planar graphs .
For this problem, we verified by an exhaustive computer search that for every . However, there are larger bipartite planar graphs for which this is not true.
Similarly, the bounds we give in Theorem 30 are tight for cubic graphs. But what about quartic, that is, 4-regular graphs? In general, we find the further study of perfect Italian domination interesting for other regular graphs.
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