Determining All Maximum Uniquely Restricted Matching in Bipartite Graphs

Determining All Maximum Uniquely Restricted Matching in Bipartite Graphs

Guohun Zhu Guilin University of Electronic Technology No.1 Jinji Road, Guilin, China, 541004
Abstract

The approach mapping from a matching of bipartite graphs to digraphs has been successfully used for forcing set problem, in this paper, it is extended to uniquely restricted matching problem. We show to determine a uniquely restricted matching in a bipartite graph is equivalent to recognition a acyclic digraph. Based on these results, it proves that determine the bipartite graphs with all maximum matching are uniquely restricted is polynomial time. This answers an open question of Levit and Mandrescu(Discrete Applied Mathematics 132(2004) 163–164).

keywords:
Bipartite graph, Directed cycle, BD-mapping, Uniquely restricted matching
journal: Information Processing Letters

1 Introduction

Let be a bipartite graph, a set of edges is a matching if no two edges of share a common vertex. A matching is uniquely restricted if its saturated vertices induce a subgraph which has a unique perfect matching and denotes as . A subset edges is a forcing set for a matching if is in no other perfect matching of . Let us denote the subgraph induced by the dedges of (laso known as the saturated vertices) as , and name all of the vetices not saturated by as free vertex set .

Maximum matching problems are well known problem in graph theory and are proved to be solved in polynomial time[2]. But many restricted maximum matching problems are NP-complete, for example, Finding the maximum is NP-complete in bipartite graphs [3], the smallest forcing set problem is also NP-complete in cubic bipartite graphs [1]

On the positive side, it is proved that the determine a matching is uniquely restricted in bipartite graph could be recognized in [3]. There exists a polynomial time algorithm to determine the if is unicycle graph[5]. And [4] also shown that unique perfect matching bipartite graph could be find in polynomial time. At the end of [4], they raised an open problem as follows:

Problem.

how to recognize that all maximum matching in a bipartite graph are uniquely restricted?

In this paper we will answer this question in two steps. Firstly, it shows a mapping from bipartite graph to digraph, and then it gives a necessary and sufficient condition on a uniquely restricted matching in bipartite graphs is equaivlent to the acyclic digraph. Secondly, it proves to determine all the maximum matching uniquely restricted or not is eqaivlent to find no more than two path between two vertices. In addition, it shows that uniquely perfect matching in bipartite graphs is as simple as recoginze the an acyclic digraph.

2 Illustration the main technology

The main technonlogy in this paper have a successful implementation on finding forcing set problem in [1]. But firstly, let us repeat the theorem in [3].

Theorem 1

[3] A matching of a graph is uniquely restricted if and only if is alternating cycle-free.

Then let us define a mapping from of a bipartite graph to a digraph and named as -mapping in this paper, this mapping is much more clearly than the definition on page 292 of [6] and definition on page 3 of [1] which denotes by .

Definition 2

Given a matching of bipartite graph , a -mapping digraph of defines as follows

(1)
(2)

It is easy to observe that follows theorem could be equivalent to the theorem 1.

Theorem 3

Let be -mapping digraph of a matching in bipartite graph with vertices, is uniquely matching in if and only if is acyclic.

{@proof}

[Proof.] Suppose that is acyclic digraph, every vertex on could be divided into a pair of vertex and become a new directed graph , which is also a acyclic digraph and without alternative cycle. Moreover, there has a matching include number of edges. Therefore is a uniquely restricted matching.

On the other hand, assume is a uniquely restricted matching but the -mapping include at least one cycle , where or when . Then according to equation 2 of definition 2, there exists and are in , or , and . Therefor, has a even cycle, this contradict to the theorem 1.

Remark 4

Theorem 3 is very similar the proposition in [1].

Proposition 5

[1] Let be a bipartite graph, is a perfect matching of , and is a focing set of if and only if is an acyclic digraph.

In fact, Theorem 3 can deduce the known results that

Corollary 6

The bipartite graph with uniquley perfect matching has a forcing matching number

{@proof}

[Proof.] Since with uniquely perfect matching is acyclic graph, the in proposition 5 is empty set.

Let us give an example to show a bipartite graph and the in Fig. to end of this section.

Figure 1: The left is bipartite graph and the right is the

3 The complexity of uniquley restricted perfect matching

[4]has proved that if and only all of local maximum stable set is a greedoid, then a bipartite graph has a unique perfect matching. However, how to recognize all of local maximum stable set are greediod is equivalent to the problem of all maximum maching are uniquely restricted according to the theorem 3.3 in [4]. This section would give a more efficient algorithm to determine the unique perfect matching.

It is easy to obervious to obtain the following theorem:

Theorem 7

A matching of a bipartite graph is perfect uniquely restricted if and only if is acyclic and .

Based on theorem an algorithm shows in Algorithm 1.

Algorithm 1 Determine unique pefect matching

Input : A bipartite graph and a matching ; Output: unique perfect maching if it has, otherwise return non unique perfect matching. begin (1).Generate a BD-mapping graph . (2) if is not perfect then (3)       return non unique perfect matching (4) else (5)   if is acyclic then return unique perfect matching (6) else return non unique perfect matching   endif end

Figure 2: with uniquely prefect matching

It is clearly to see that the example in Fig. is a uniquly restricted perfect matching. Now let us consider the Fig. again. It is clearly that G in Fig. is contains a unique maximum matching . Notice that also contains a maximum matching is not uniquely restricted. Thus the theorem 3 is only necessary condition for all maximum matching are uniquely restricted.

4 Determine all uniquely restricted maximum matching in polynomial time

In first glance, greedy algorithms can apply into determing all uniquely restricted maximum matching by remove the node with degree . Unfortunately, the worst case could be exponent.

For example, let consider the bipartite graph in Fig..

Figure 3: Not all maximum matching are uniquely restricted, even remove edge , there exists and .

It is need to remove edges , then any maximum matching is not uniquely restricted. But for another example in Fig 4,

Figure 4: Any matching of is not uniquely restricted

It only remove edges , then any maximum matching is not uniquely restricted. Therefore, greedy algorithms on removing vertex with degree could not efficiently to determing all maximum matching are unqiuely restricted.

However, a uniquley restricted maximum matching will include vertex with degree , let us define a matching is a greedy matching if the free vertex set of do not include vertex with degree or all edges of saturate a vertex degree . Then let us define an extend BD-mapping digraph, which consist of the free vertex set and a BD-mapping digraph , where is greedy matching.

Definition 8

An extend BD-mapping digraph of bipartitie graphs with a greedy matching is follows. and , where is a RZ-mapping digraph and is a pair of arcs between (), and all of if

An example of extend is shown in Fig.. Now let we give a necessary and sufficient condition of all maximum matching are uniquely restricted.

Figure 5: Extend BD-mapping from a matching of graph to a digraph
Theorem 9

Let response to the extends BD-mapping digraph of a bipartite graph with uniquely restricted maximum matching and and are the set of terminal nodes or start nodes in , when satisfies following one of three conditions

c1.

all of , there exists only one path from to .

c2.

all of , there exits only one path from to .

c3.

for any two , if there exists at most one have the path from to and to , (or conversly, there exists at most one have the path from to and to .

then all maximum matching of are uniquely restricted, where is set of free nodes of .

{@proof}

[Proof.]

c1.

If a free node have more than two disjoint paths , to , then it implies that two consecustive edges , not belong to and in .

since terminal node in is always respect to the vertex with degree . There exits two disjoint path to , which implies that two consecuistive edges , belongs o and also in . Therefor there exists at least 4 edges in a cycle and not in ,if we remove the degree 1 node from , minus , but it can plus by extends the and .

c2.

The same principle for the start node if is the source node of .

c3.

Let us prove it by constraction, suppose there exits a node have the path from to and from to , also there exits a node have the path from to and from to . Then there exits a path from to is length of ( respectively), but the nodes in the is ( respectively), therefor, there exists a cycle in length of , which have a matching length of .

According to the theorem 9, it is easy to design a deterministic all maximum matching restrict or not in polynomial algorithm since deterministic free node to the terminal node or start node have more than two disjoint path is clearly in polynomial time. The Algorithm shows the algorithms for determine all maximum matching restricted.

Algorithm 2 Determine all maximum matching uniquely restricted

Input : A bipartite graph with a greedy matching ; Output: return true if all maximum matching are uniquely restricted, otherwise return false. begin (1).Generate a BD-mapping graph . (2) if is not perfect then (3)       return non unique perfect matching (4) else (5)   if is acyclic then return unique perfect matching (6) else return non unique perfect matching   endif end

5 Discussion

The mapping from a matching of bipartite graph to digraph had been succesful solve forcing matching problem in bipartite graph of [1][6]. This paper extends it and use to solve the uniquley restricted maximum matching problem. According to the theorem 9, the open question appear in [4] to recognize the all uniquely restricted maximum matching bipartite graphs is solved in polynomial time.

References

References

  • [1] P. Adams, M. Mahdian and E.S. Mahmoodian, On the forced matching number of graphs, Discrete Math. 281 (2004), 1–12.
  • [2] H.N. Gabow, An efficient implementation of Edmonds? algorithm for maximum matching on graphs, J. Asc. Com. Mach. 23(1976), 221?-234.
  • [3] M.C. Golumbic, T. Hirst, M.Lewenstein. Uniquely restricted matchings, Algorithmica 31(2001), 139–154
  • [4] V.E. Levit, E. Mandrescu, Local maximum stable sets in bipartite graphs with uniquely restricted maximum matchings, Discrete Appl. Math. 132 (2003), 163-?174.
  • [5] V.E. Levit, E. Mandrescu, Unicycle graphs and uniquely restricted maximum matchings, Electronic Notes in Discrete Mathematics 22 (2005), 261–265
  • [6] L. Pachter and P. Kim, Forcing matchings on square grids, Discrete Math. 190 (1998) 287–294.
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