A theta in a graph is an induced subgraph consisting of two nonadjacent vertices joined by three disjoint paths. A prism in a graph is an induced subgraph consisting of two disjoint triangles joined by three disjoint paths. This paper gives a polynomial-time algorithm to test whether a graph has an induced subgraph that is either a prism or a theta.

A classic theorem of Euclidean geometry asserts that any noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal conjectured that this holds for an arbitrary finite metric space, with a certain natural definition of lines in a metric space.We prove that in any metric space with n points, either there is a line containing all the points or there are at least Ω( √ n) lines. This is the first polynomial lower bound on the number of lines in general finite metric spaces. In the more general setting of pseudometric betweenness, we prove a corresponding bound of Ω(n 2/5 ) lines. When the metric space is induced by a connected graph, we prove that either there is a line containing all the points or there are Ω(n 4/7 ) lines, improving the previous Ω(n 2/7 ) bound. We also prove that the number of lines in an n-point metric space is at least n/5w, where w is the number of different distances in the space, and we give an Ω(n 4/3 ) lower bound on the number of lines in metric spaces induced by graphs with constant diameter, as well as spaces where all the positive distances are from {1, 2, 3}.

A classical theorem of Euclidean geometry asserts that any noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal conjectured a generalization of this result to arbitrary finite metric spaces, with a particular definition of lines in a metric space. We prove it for metric spaces induced by connected distance-hereditary graphs -a graph G is called distance-hereditary if the distance between two vertices u and v in any connected induced subgraph H of G is equal to the distance between u and v in G.

We give polynomial-time randomized algorithms for computing the girth and the cogirth of binary matroids that are low-rank perturbations of graphic matroids. a cycle of M (A) * . So the cogirth of M (A) is the size of the smallest non-empty cocycle. Again, for this paper, this is the most convenient way to view cogirth.Motivation. The problem of computing the girth of a binary matroid has received a lot of attention due to its well-known connection with coding theory. If A is the parity-check matrix of a binary linear code C, then the distance of C is equal to the girth of the binary matroid M (A). In a landmark paper, Vardy [13] proved that the problem of computing girth in binary matroids is N P-hard. On the other hand, there are significant classes of binary matroids in which one can efficiently compute girth; for example, the class of graphic matroids and the class of cographic matroids. Geelen, Gerards, and Whittle [3] posed the following conjecture. Conjecture 1.3. For any proper minor-closed class M of binary matroids, there is a polynomial-time algorithm for computing the girth of matroids in M.Here a minor-closed class of binary matroids is called proper if it does not, up to isomorphism, contain all binary matroids. In the same paper, Geelen, Gerards, and Whittle announced (without proof) the following result:Theorem 1.4. For each proper minor-closed class M of binary matroids, there exist non-negative integers k and t such that, for each vertically k-connected matroid M ∈ M, there exist matrices A, P ∈ GF(2) r×n such that A is the incidence matrix of a graph, rank(P ) ≤ t, and either M = M (A + P ) or M = M (A + P ) * .In light of this result, our Theorems 1.1 and 1.2 give significant support to Conjecture 1.3; their main shortcomings being: (1) they only apply to sufficiently connected matroids in a minor-closed class, and (2) they only give randomized algorithms.Related work. Barahona and Conforti [1] studied both the girth and cogirth problems for the class of "even-cycle matroids". Let M 1 and M 2 be binary matroids on the same ground set. We call M 1 a rank-t perturbation of M 2 if M 1 has a representation A and M 2 has a representation B such that B − A has rank t. We call M 1 an even-cycle matroid if it is a rank-1 perturbation of a graphic matroid M 2 and r(M 1 ) = r(M 2 ) + 1.Barahona and Conforti gave an efficient deterministic algorithm for computing girth of even-cycle matroids. They also noted that the problem of computing cogirth for these matroids is closely related to the

Abstract. For a set of matroids M, let ex M (n) be the maximum size of a simple rank-n matroid in M. We prove that, for any finite field F, if M is a minor-closed class of F-representable matroids of bounded branch-width, then lim n→∞ ex M (n)/n exists and is a rational number, ∆. We also show that ex M (n) − ∆n is periodic when n is sufficiently large and that ex M is achieved by a subclass of M of bounded path-width.