A word-representable graph is a simple graph which can be represented by a word over the vertices of such that any two vertices are adjacent in if and only if they alternate in . While the origin of word-representable graphs lies in the study of Perkins semigroups , this notion covers many important classes of graphs including comparability graphs, circle graphs and 3-colorable graphs. The literature has several important contributions to the theory of word-representable graphs and their connections to various concepts. A comprehensive introduction to word-representable graphs and their connections and contributions can be found in the monograph .
The representation number of a word-representable graph is the minimum number such that the graph is represented by a word containing exactly copies of each letter. The class of complete graphs is the class of graphs with representation number 1 and it was established in  that the class of circle graphs characterizes the graphs with representation number 2. Although there are examples of graphs with higher representation number, no characterization of graphs is available for other numbers.
The class of comparability graphs – the graphs which admit a transitive orientation – play a vital role in the theory of word-representable graphs. In fact, the neighborhood of each vertex in a word-representable graph is a comparability graph . Furthermore, the class of comparability graphs is precisely the class of permutationally representable graphs, that is, they can be represented by a concatenation of permutations of their vertices . We call the minimum number of such permutations of vertices of a comparability graph its permutation representation number. Every comparability graph corresponds to a partially ordered set (poset) based on one of its transitive orientations. Due to their correspondence with posets, comparability graphs received importance in the literature. It is known that finding the permutation representation number of a comparability graph is equivalent to finding the dimension of the corresponding poset . However, the problem of finding the dimension of a poset is NP-hard .
The class of bipartite graphs is a subclass of the class of comparability graphs. While it is an open problem to determine the representation number of a comparability graph, it was conjectured in  that the representation number of a bipartite graph on vertices is at most . In this paper, we propose a polynomial time relabeling algorithm to produce a word that is a concatenation of permutations of vertices representing a given bipartite graph. Thus we obtain an upper bound for the permutation representation number of bipartite graphs and consequently an upper bound for their representation number. This indeed gives us an upper bound for the dimension of the posets corresponding to bipartite graphs. In fact, the upper bound for the permutation representation number is tight for the class of bipartite graphs.
Let be a finite set. A word over is a finite sequence of elements of . The words are written by juxtaposing the symbols of the sequence. The empty sequence, called the empty word, is denoted by . A subword of a word is a subsequence of the sequence and it is denoted by . For instance, . Clearly, is a partial order relation on the set of all words over . We say the letters and alternate in the word if the subword consisting of all occurrences of and , that is the subword obtained by deleting all other letters of , is of the form or
(of even or odd length). A wordis said to be -uniform if the number of occurrences of each letter in is .
Let be a set and . We write to represent the word in which the symbols of appear exactly once and are arranged such that the subscripts are in decreasing order.
In this paper, the term graph represents only a simple and undirected graph, i.e., a graph without loops or parallel edges. To distinguish between two-letter words and edges of graphs, we write to denote an undirected edge between vertices and . A directed edge from to is denoted by . The neighborhood of a vertex in a graph, denoted by , is the set of all vertices adjacent to in the graph. An orientation of a graph is an assignment of direction to each edge so that the resulting graph is a directed graph. An orientation of a graph is said to be a transitive orientation, if the adjacency relation on the vertices in the resulting directed graph is transitive. A graph is said to be a bipartite graph if its vertex set can be partitioned into , called bipartition, such that every edge in connects a vertex in to a vertex in . A bipartite graph with bipartition , where and , is said to be complete, denoted by , if every vertex in is adjacent to all vertices of .
A graph is said to be word-representable if there is a word over the set such that two vertices and are adjacent in if and only if and alternate in . A graph is said to be -word-representable if there is -uniform word representing the graph. Every word-representable graph is -word-representable for some . The representation number of a word-representable graph , denoted by , is the smallest number such that is -word-representable. A graph is said to be permutationally representable if it can be represented by a word of the form , where each is a permutation of vertices of . Furthermore, if a graph is permutationally representable involving permutations, then the graph is called permutationally -representable.
Let be a permutationally representable graph. The permutation representation number of , denoted by , is the smallest such that is permutationally -representable.
If is a permutationally representable graph, then .
The class of complete graphs has permutation representation number 1, i.e., for all , where is the complete graph on vertices. It was shown in [2, see the proof of Theorem 4] that (for ) and in  that (for ), where is the graph, called a crown graph, obtained by removing a perfect matching in . Further, it was conjectured in [1, see Conjecture 1 in page 93] that every bipartite graph on vertices has representation number at most .
A graph is said to be a comparability graph if it admits a transitive orientation. Every comparability graph is word-representable. In fact, the class of comparability graphs is precisely the class of permutationally representable graphs . Every bipartite graph is a comparability graph. Based on its transitive relation, every comparability graph corresponds to a partially ordered set (poset). In fact, the class of bipartite graphs is precisely the class of comparability graphs that are isomorphic to the Hasse diagrams of the corresponding posets.
Let be a comparability graph and be the Hasse diagram of the corresponding poset. If the graphs and are isomorphic, then is a bipartite graph, and vice versa.
The dimension of a poset is the minimum number of linear orders of its elements such that their intersection is the partial order of the poset. It was shown in  that a comparability graph is permutationally -representable if and only if the poset corresponding to has dimension at most . However, since the problem of finding the dimension of a poset is NP-hard , finding the permutation representation number of a comparability graph is NP-hard. Hence, in the direction of finding the permutation representation number of some subclasses of comparability graphs, in , the authors obtained the permutation representation number of crown graph . Here, the authors constructed linear orders whose intersection happens to be the corresponding poset of . In the following section, we extend the idea to arbitrary bipartite graphs and construct linear orders of the elements of corresponding poset.
3 Main Result
In this section, we devise an algorithmic procedure to construct a word representing permutationally a given bipartite graph. Subsequently, we obtain an upper bound for the permutation representation number of bipartite graphs. Consequently, in view of the correspondence between comparability graphs and posets, we also present an upper bound for the dimension of a special class of posets which correspond to bipartite graphs.
The permutation representation number of a graph does not change by the inclusion or deletion of isolated vertices. If a graph (with at least 3 vertices) has isolated vertices, say , then consider the induced subgraph of on the vertex set . That is, is obtained by deleting the vertices from . Note that is permutationally representable if and only if is permutationally representable. In fact, if , , …, are permutations of vertices of such that the word represents the graph permutationally, then we can see that is also permutationally -representable. For instance, the following word with permutations on the vertices of represents permutationally:
where and , the reversal of .
In what follows, always denotes a bipartite graph with bipartition and . In view of Remark 3, we assume that has no isolated vertices.
3.1 Relabeling algorithm
We now present an algorithm to determine permutations of vertices of a given bipartite graph such that their concatenation represents . It is achieved by relabeling the vertices of in a particular order based on their nonadjacent vertices and construct certain permutations of the vertices of . The algorithm is presented in the following.
Given a bipartite graph , suppose and .
If for all , then consider the following word and exit:
Else, choose in such that and label it as .
Relabel the remaining vertices of as , arbitrarily.
Relabel the vertices in as follows:
Set ; .
For to ,
Let , say .
Relabel the vertices of as , arbitrarily.
Set ; .
If , then relabel the remaining vertices of as , arbitrarily.
Create a list of permutations of the vertices as per the following:
For to ,
If , then set
else, set .
If for some then set
else, set .
Report the word obtained by replacing the original labels of the vertices of in the word .
The relabeling algorithm runs in a polynomial time. In fact, its complexity is . Since the relabeling algorithm works by identifying the set of nonadjacent vertices (adjacent vertices) for each vertex of , so the only possible nonadjacent vertices would be the vertices from .
We demonstrate the relabeling algorithm on the bipartite graph given in Figure 2. Although the selection of vertices can be arbitrary, here we implemented the algorithm based on their degrees. Accordingly, the vertices of , viz., are relabeled, respectively, as . Further, the vertices of , viz., are relabeled as , respectively. The relabeled graph is presented in Figure 2. As per the algorithm, the permutations and of vertices of are produced below. Note that, as , and further will be nonempty.
Concatenating all these permutations we get
which represents . Relabeling the vertices in with their original labels we get
which represents .
3.2 Correctness of the algorithm
In the following theorem we establish the correctness of the relabeling algorithm.
The word generated by the relabeling algorithm permutationally represents the bipartite graph .
If is a complete bipartite graph, then the algorithm produces the word given in Step 3. It is a routine verification to observe that the word represents . If is not a complete bipartite graph, let be the graph obtained after relabeling in Step 6. We show that the word represents . Accordingly, it is straightforward that represents .
We now prove that is an edge in if and only if and alternate in the word .
Suppose is an edge in . Without loss of generality assume and . If then clearly (when ) for all . Further, since we have too. Hence, and alternate in . If for , then and also (when ). Further (when ). Hence, for all nonempty so and alternate in .
Conversely, suppose is not an edge in . We deal with this part in three cases, as the case and is symmetric to Case 2. In each case we identify two permutations produced by the algorithm: that one with the subword and the other with the subword .
Case 1: . Let and with . Note . Clearly . If , then and . Otherwise, so that and .
Case 2: and .
Subcase 2.1: . Since , . Since there are no isolated vertices, and hence there exists such that . Depending on or not, or , respectively. Accordingly, or .
Subcase 2.2: for some . Clearly, . Since , we have so that . Then .
Case 3: . Let and with . Clearly, . As there are no isolated vertices, . Hence, there exists such that .
If , then again and .
If , let and note that . If , then . Otherwise, so that .
Hence, and do not alternate in . This completes the proof.
3.3 Upper bounds
We now obtain an upper bound for the permutation representation number of arbitrary bipartite graphs and also an improved upper bound in case of bipartite graphs with a special property.
First note that, for each vertex in , the relabeling algorithm produces at most one permutation of the vertices of . Hence, the algorithm produces a word consisting of at most permutations of vertices of . Accordingly, we obtain an upper bound for the representation number of bipartite graphs in the following corollary of Theorem 3.1.
For a bipartite graph , the permutation representation number . Consequently, .
Since (for ), the upper bound obtained in Corollary 1 is tight for the permutation representation number of the class of bipartite graphs.
The dimension of a poset corresponding to a bipartite graph is bounded above by .
Considering the possibility of occurrence of the empty word in Step 6(b) of the relabeling algorithm, the upper bound given in Corollary 1 can be improved for a special type of bipartite graphs as per the following corollary.
Let be the set of vertices each of which is adjacent to all vertices of in a bipartite graph , then .
In this paper, we obtained a word that is a concatenation of permutations of the vertices of a given bipartite graph through a polynomial time algorithm such that represent . Accordingly, we obtain a tight upper bound for the permutation representation number of bipartite graphs and hence an upper bound for the representation number of bipartite graphs. We believe that the relabeling algorithm introduced in this paper has a scope to extend to a larger class of comparability graphs.
We are thankful to the referees for their comments which improved the presentation of the paper. The first author is thankful to the Council of Scientific and Industrial Research (CSIR), Government of India, for awarding the research fellowship for pursuing Ph.D. at IIT Guwahati.
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