_{1}

^{*}

The bipartite
Star
_{123}-free graphs were introduced by V. Lozin in [1] to generalize some already known classes of bipartite graphs. In this paper, we extend to bipartite
Star
_{123}-free graphs a linear time algorithm of J. L. Fouquet, V. Giakoumakis and J. M. Vanherpe for finding a maximum matching in bipartite
Star
_{123},
P
_{7}-free graphs presented in [2]. Our algorithm is a solution of Lozin’s conjecture.

A matching M of a graph

and Paul gave an

For terms not defined in the paper the reader can refer to [

Definition 1 [

Property 2 [

Such partition is referred as associated partition of G and is denoted by the ordered pair (

Property 3 [

1)

2)

The partition

From

associated to this decomposition. The internal nodes are labeled according to the type of decomposition applied, while every leaf correspond to a vertex of G. Hence there are four types of internal nodes, parallel node (labeled P), series node (labeled S),

Lozin in [_{123}-free graphs.

Theorem 4 [_{123}-free graph. One of the following hold.

1) G is

2) G and

3) The representative graph of G or the bi-complement of the representative graph of G is a path

It is shown in [

Definition 5 [

G into a monochromatic sets

Definition 6 [

G into a monochromatic sets

The construction of the canonical decomposition tree of a bipartite Star_{123}-free graph can be obtained in linear time from the algorithm given by Quaddoura in [_{123}-free graph and its canonical decomposition tree.

In this section we will extend the techniques developed in [_{123}-free graph. We present first the required tools for this purpose.

A classical tool for solving the maximum matching problem was introduced by Berge in [

Theorem 7 [

Consider a bipartite graph G such that G admit a decomposition according to some rule into two graphs

Let

Let U be the set of M-unsaturated vertices, a Split operation occurs if there exists an edge of M say xy and vertices u and v belonging to U such that u is adjacent to x and v is adjacent to y. In that case the Split operation constructs a new matching

Let now G be a bipartite Star_{123}-free graph and

Consider now the set

Let

Let now U be the set of M-unsaturated vertices, and Let

Theorem 8 [

Theorem 9 [

In this section we will develop an

Procedure MAXMATCH

Procedure MAXMATCH EP_{k}, EC_{k}

1)

2) if

3) for

begin for

4)

5)

end for

Theorem 10. Let

Proof. Let

Claim 1. There is no black vertex of P in

Proof. Let

Claim 2. There is no white vertex of P in

Proof. Let

Suppose that G is an

Suppose now G is an

The following

Note that the matching obtained by the Procedure MAXMATCH

Recall that when an edge xy is added to a matching M by a Match operation

Lemma 11. Let

i | M | ||
---|---|---|---|

1 | ____ | ||

2 | |||

3 | ____ | ||

4 | ____ |

Proof. By the hypothesis of the Lemma, all the M-unsaturated vertices must be in independent sets. Obviously any three consecutive sets are independent and the maximum number of independent sets is three. □

Assume that

Procedure M-unsaturated vertices (G, M)

1) Find the small index

2) if there is no such s then return M is maximum

else

3) if

//when

4) if

5) else return

else //

6) if

7) else if

8) else return

According to Lemma 11, one of the two M-unsaturated vertices of any M-augmenting path in G is in

To augment the size of M, Split operations can be done between the M-unsaturated vertices of

Procedure Split (M, V_{s}, V_{s}_{+1})

1) if

2) else

3) while

Begin while

4) let

5)

// assuming that u and x also v and y are of different color

end while

The following Lemma describes the structure of a M-augmenting path whose extremities belong to

Lemma 12. After the execution of Procedure Split

・

・ P can be reduced to a M-augmenting path

Proof. Let

Let

Let

Consider now a M-augmenting path in G such that its M-unsaturated vertices are in

Procedure Split (M, V_{s}_{+1}, V_{s}_{+2})

1) if

2) else

3) while

begin while

4) let

5)

//assuming that u and x also v and y are of different color

end while

Lemma 13. After the execution of Procedure Split

・

・ P can be reduced to a M-augmenting path

We start now by developing a Procedure for a maximum matching in

Procedure MATCH (G)

1)

2)

3) For

begin for

4)

5) while

begin while

6)

7) if

end while

8)

9)

10) while

begin while

11)

12) if

end while

13)

end for

Procedure MATCH (G) works as following, for every

・ Add to M the possible edges between

・ Add to M the possible edges between

Observation 14. According to Procedure MATCH (G):

・ if

・ if

・ if

The following

The combination of Procedures MATCH (G), M-unsaturated vertices

a little addition. Theorem 15 proves their correctness.

Procedure MAXMATCH

1) MATCH

2) M-unsaturated vertices

3) Split

4) Split

Procedure MAXMATCH

1) MATCH

2) M-unsaturated vertices

3) Split

4) Split

5) if

//Assuming that x and the vertices of

6)

7)

8) while

begin while

let

9)

10)

end while

Theorem 15. Procedure MAXMATCH

Proof. Suppose that after execution of Procedure MAXMATCH

Let

Claim 1. The edge

Proof. Suppose that the edge

i | h | l | j | j | M | ||
---|---|---|---|---|---|---|---|

2 | 2 | 2 | 2 | ____ | 1 | ||

2 | ____ | 3 | ____ | ||||

3 | 2 | 3 | 2 | 3 | |||

4 | ____ | 3 | ____ | ||||

4 | 4 | 3 | 4 | ____ | 3 | ||

6 | ____ | 6 | ____ |

tained by

exists and must precedes the operation

edge

Claim 2.

Proof. If

Since

Assume that

Assume finally that

Claim 3. The edge

Proof. Suppose that the edge

Claim 4.

Proof. If

Since

then

Lets apply the Procedure MAXMATCH

Procedure MATCH (G) produces the matching

turated vertices

Since

Let us present now our algorithm for the maximum matching problem on bipartite

Input: A bipartite

Output: M a maximum matching of G and U the set of M-unsaturated vertices of G.

1) Let

2) If

3) Else if

4) If

vertices.

5) Else if

6) Else

7) Replace

8) Else let

9) Let

10)

11) If

12) Else if

13) Else

We show now that the complexity of our algorithm is

The total number of Match operations performed by

Consider the steps 5 and 6 which are the Procedures MAXMATCH

is

Since the size of the matching obtained by MATCH (G) is less than or equal to

The total number of Match or Split operations performed in steps 8 to 13 is bounded by the size of maximum matching obtained, which is less or equal to

Finally, since the number of visited nodes in

The maximum matching is computed in

This research is funded by the Deanship of Research and Graduate Studies in Zarqa University/Jordan. The author is grateful to anonymous referee’s suggestion and improvement of the presentation of this paper.

RuzaynQuaddoura, (2016) Solving the Maximum Matching Problem on Bipartite Star_{123}-Free Graphs in Linear Time. Open Journal of Discrete Mathematics,06,13-24. doi: 10.4236/ojdm.2016.61003