On Critical Circuits in k-Connected Matroids (original) (raw)

On elements in small cocircuits in minimally k-connected graphs and matroids

Discrete Mathematics, 2002

We give a lower bound on the number of edges meeting some vertex of degree k in terms of the total number of edges in a minimally k-connected graph. This lower bound is tight if k is two or three. The extremal graphs in the case that k = 2 are characterized. We also give a lower bound on the number of elements meeting some 2-element cocircuit in terms of the total number of elements in a minimally 2-connected matroid. This lower bound is tight and the extremal matroids are characterized.

On the circuit-spectrum of binary matroids

2011

Murty, in 1971, characterized the connected binary matroids with all circuits having the same size. We characterize the connected binary matroids with circuits of two different sizes, where the largest size is odd. As a consequence of this result we obtain both Murty's result and other results on binary matroids with circuits of only two sizes. We also show that it will be difficult to complete the general case of this problem.

Removing circuits in 3-connected binary matroids

Discrete Mathematics, 2009

For a k-connected graph or matroid M, where k is a fixed positive integer, we say that a subset X of E(M) is k-removable provided M\X is k-connected. In this paper, we obtain a sharp condition on the size of a 3-connected binary matroid to have a 3-removable circuit.

Circuit and cocircuit partitions of binary matroids

Czechoslovak Mathematical Journal, 2006

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Infinite subgraphs as matroid circuits

Journal of Combinatorial Theory, Series B, 1979

A matroidal family V is defined to be a collection of graphs such that, for any given graph G, the subgraphs of G isomorphic to a graph in V satisfy the matroid circuit-axioms. Here matroidal families closed under homeomorphism are considered. A theorem of Simks-Pereira shows that when only finite connected graphs are allowed as members of Q, two matroids arise: the cycle matroid and bicircular matroid. Here this theorem is generalized in two directions: the graphs are allowed to be infinite, and they are allowed to be disconnected. In the first case four structures result and in the second case two infinite families of rnatroids are obtained. The main theorem concerns the structures resulting when both restrictions are relaxed simultaneously. We will use standard graph theory terminology as far as possible, as found in [l], [2], or [13]. All graphs will be undirected and possibly infinite, and loops and multiple edges will be allowed. If G is a graph, E(G) denotes the set of edges of G and G\e denotes the graph obtained from G by deleting the edge e. A graph H is homeomorphic from G if it is isomorphic to a graph obtained from G by replacing each edge by a finite path and a graph K is homeomorphic to G if there exists some graph H such that G and K are both homeomorphic from H. The matroid theory terminology will follow [12]. One of the many ways to define a matroid on a finite set is by means of its collection % of circuits, which satisfies the following two axioms: (Cl) No member of V properly contains another.

On chains of 3-connected matroids

Discrete Applied Mathematics, 1986

A sequence of k-connected matroids No, Ni ..... Nm is called a k- chain, from No to N,,, if Ni_ I is a minor of N t (i= i ..... m); this chain is said to have gap t=max {IE(N,)I-IE(N~-t)I: i= 1, ...,m}. Chains of gap 1 are said to be dense.

On the Structure of 3-connected Matroids and Graphs

European Journal of Combinatorics, 2000

An element e of a 3-connected matroid M is essential if neither the deletion M\e nor the contraction M/e is 3-connected. Tutte's Wheels and Whirls Theorem proves that the only 3-connected matroids in which every element is essential are the wheels and whirls. In this paper, we consider those 3-connected matroids that have some non-essential elements, showing that every such matroid M must have at least two such elements. We prove that the essential elements of M can be partitioned into classes where two elements are in the same class if M has a fan, a maximal partial wheel, containing both. We also prove that if an essential element e of M is in more than one fan, then that fan has three or five elements; in the latter case, e is in exactly three fans. Moreover, we show that if M has a fan with 2k or 2k + 1 elements for some k ≥ 2, then M can be obtained by sticking together a (k + 1)-spoked wheel and a certain 3-connected minor of M. The results proved here will be used elsewhere to completely determine all 3-connected matroids with exactly two non-essential elements.

Connectivity Properties of Matroids

The bases-exchange graph of a matroid is the graph whose vertices are the bases of the matroid, and two bases are connected by an edge if and only if one can be obtained from the other by the exchange of a single pair of elements. In this paper we prove that a matroid is \connected" if and only if the \restricted bases-exchange graph" (the bases-exchange graph restricted to exchanges involving only one speci c element e) is connected. This provides an alternative de nition of matroid connectivity. Moreover, it shows that the connected components of the restricted bases-exchange graph satisfy a \ratios-condition", namely, that the ratio of the number of bases containing e to the number of bases not containing e is the same for each connected component of the restricted bases-exchange graph. We further show that if a more general ratios-condition is also true, namely, that any fraction of the bases containing e is adjacent to at least a fraction of the bases not containing e (where is any real number between 0 and 1), then the bases-exchange graph has the following expansion property : \For any bipartition of its vertices, the number of edges incident to both partition classes is at least as large as the size of the smaller partition". In fact, this was our original motivation for studying matroid connectivity, since such an expansion property yields e cient randomized approximation algorithms to count the number of bases of a matroid 18].

A proof of Connelly's conjecture on 3-connected circuits of the rigidity matroid

Journal of Combinatorial Theory, 2003

A graph G ¼ ðV ; EÞ is called a generic circuit if jEj ¼ 2jV j À 2 and every X CV with 2pjX jpjV j À 1 satisfies iðX Þp2jX j À 3: Here iðX Þ denotes the number of edges induced by X : The operation extension subdivides an edge uw of a graph by a new vertex v and adds a new edge vz for some vertex zau; w: Connelly conjectured that every 3-connected generic circuit can be obtained from K 4 by a sequence of extensions. We prove this conjecture. As a corollary, we also obtain a special case of a conjecture of Hendrickson on generically globally rigid graphs. r

Non-Separating Cocircuits and Graphicness in Matroids

Cornell University - arXiv, 2012

Let M be a 3-connected binary matroid and let Y (M) be the set of elements of M avoiding at least r (M) + 1 non-separating cocircuits of M. Lemos proved that M is non-graphic if and only if Y (M) =. We generalize this result when by establishing that Y (M) is very large when M is non-graphic and M has no M * (K ′′′ 3,3)-minor if M is regular. More precisely that |E (M)− Y (M)| ≤ 1 in this case. We conjecture that when M is a regular matroid with an M * (K 3,3)-minor, then r * M (E (M) − Y (M)) ≤ 2. The proof of such conjecture is reduced to a computational verification.