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Volume 2011 (2011), Article ID 208478, 4 pages
A Characterization of Uniform Matroids
Sciences College, Al Imam Muhammad Ibn Saud Islamic University, P.O. Box 286574, Riyadh 11323, Saudi Arabia
Received 18 September 2011; Accepted 23 October 2011
Academic Editor: A. Jaballah
Copyright © 2011 Brahim Chaourar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper gives a characterization of uniform matroids by means of locked subsets. Locked subsets are 2-connected subsets, their complements are 2-connected in the dual, and the minimum rank of both is 2. Locked subsets give the nontrivial facets of the bases polytope.
Let be a finite set, and let be a matroid defined on . If is 2-connected, then we will say that a proper subset of ; that is, , is locked if is nonseparable or 2-connected in is nonseparable or 2-connected in and or . Observe that is locked in a matroid if and only if and are both connected and . Locked subsets give the nontrivial facets of the bases polytope. We will denote the class of these subsets by or . If is not 2-connected, then where each is 2-connected, , and the class of locked subsets is the union of such classes in each 2-connected component involved in the direct sums. Locked subsets were introduced by Chaourar ([3–5]) to describe some facets of the cone and the polytope generated by the matroid bases. We will denote the 2-sum of two matroids and using the basepoint by or if there is no confusion. Since 2-sums with and are, respectively, identity and deletion, then we will consider only proper 2-sums without nor .
Let be the class of matroids obtained by means of 1-sums (or direct sums) and (proper) 2-sums of uniform matroids together with all minors of such matroids. Since and (see ), then is closed under the taking of duals. It is also clear that is closed under the taking of minors.
If is not a 3-connected matroid, then, using a theorem of Oxley (see ), can be constructed from 3-connected minors of it by a sequence of the operations of 1-sum and 2-sum.
The purpose of this paper is to characterize uniform matroids by means of locked subsets. There are exactly five 3-connected matroids of rank 3 on a 6-element set. These matroids can be obtained from by relaxing zero, one, two, three, or four circuit-hyperplanes. The matroids are, respectively, , the rank-3 whirl , and the uniform matroid (see ).
The remaining of the paper is organized as follows: in Section 2, we will give a characterization of uniform matroids by means of locked subsets, two consequences are given in Section 3, and the conclusion is given is Section 4.
2. The Characterization
We will need to three lemmas in this section.
Lemma 2.1. .
Proof. Direct from the definition of a locked subset.
Lemma 2.2. Let be a 3-connected minor of a 2-connected matroid . If , then .
Proof. Using duality and Lemma 2.1, it suffices to prove that, if , then .
Suppose that is locked in . We establish that , when , or , when , is locked in . If spans in , then is connected because is not a loop of . As is connected, it follows that is locked in . If does not span in , then is not a loop of . Therefore, is connected because is connected (remember that is locked in ). Thus, is locked in .
The following last lemma of this section was proved by Walton  and we give here a new proof based on locked subsets.
Lemma 2.3. Let be a 3-connected matroid having no isomorphic minor to any of , and . Then M is uniform.
Proof. Suppose by contradiction that is not uniform. It follows that there exists a subset of such that and contains a circuit . Without loss of generality, we can suppose that because, if it is not, we can contract some elements of keeping as a circuit and decreasing its cardinality. Now we delete all elements of . Let be the obtained matroid.
Case 1. If , then let be a base of included into . If , then contract some elements of keeping as a base and as a circuit with . is a locked subset of the matroid because is 2-connected, is a cocircuit, and . Thus, is one of the excluded minors, a contradiction.Case 2. If , then is a series extension of a uniform matroid. By induction on , the matroid , obtained by contracting one element in the series closure , is uniform. But intersect so there are two parallel elements and in . Since , then , a contradiction.
Here we give our main result.
Theorem 2.4. If is a 3-connected matroid, then the following assertions are equivalent:(i) is a uniform matroid,(ii).
Proof. (i)(ii) Using the fact that there is a unique closed and 2-connected subset which is .
(ii)(i) Using Lemma 2.2, any minor of verifies . So has no isomorphic minor to any of , and , because any of these excluded minors has at least one locked subset (circuit of rank 3). By Lemma 2.3, is uniform.
Note that (i) implies (ii), in Theorem 2.4, even if is not 3-connected.
3. Some Consequences
We will give two corollaries of our characterization.
The first one is a characterization by excluded minors and is almost a restatement of Lemma 2.3, and Walton should be credited for this result:
Corollary 3.1. The following assertions are equivalent for a matroid :(i) is a minor of 1-sums and 2-sums of uniform matroids,(ii) has no isomorphic to any of and .
Proof. (i)(ii) By contradiction, suppose that has one isomorphic to any of the excluded minors. Since all the excluded minors are 3-connected, then at least one of the 3-connected components used to construct by means of 1-sums and 2-sums has one such excluded minor. Let be this excluded minor. Since the number of locked subsets for any excluded minor is at least 1, then, using Lemmas 2.2 and 2.3 and Theorem 2.4, and is not uniform.
(ii)(i) If is 3-connected, then, by Lemma 2.3, is uniform. If is not 3-connected, then can be construct using 3-connected matroids by means of 1-sum and 2-sum. It follows that no one of these matroids has an isomorphic to any of the excluded minors and, by Lemma 2.3, all these matroids are uniform.
We will need the following result of Chaourar  to deduce the second corollary.
Theorem 3.2. If is a 2-connected matroid, then its bases’ polytope is given by the following constraints:
We have given a characterization of uniform matroids by means of locked subsets and two consequences of this characterization.
Many thanks are due to Al Imam Muhammad Bin Saud University for financially supporting this research under the Grant no. 301216.
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