We show that, in any undirected graph, splitting-off can be performed while preserving all cuts of value at most 4/3 times the minimum value, and this is the best possible. This generalizes a classical splitting-off result of Lovász.

We show that under reasonable conditions, the spines of the handlebodies of a strongly irreducible Heegaard splitting will intersect a closed ball in a graph which is isotopic into the boundary of the ball. This is in some sense a generalization of the results by Scharlemann on how a strongly irreducible Heegaard splitting surface can intersect a ball. AMS Classification 57M27; 57Q10

Given a class of groups C, a group G is strongly accessible over C if there is a bound on the number of terms in a sequence Λ1, Λ2, . . . , Λn of graph of groups decompositions of G with edge groups in C such that Λ1 is the trivial decomposition (with 1-vertex) and for i > 1, Λi is obtained from Λi−1 by non-trivially and compatibly splitting a vertex group of Λi−1 over a group in C, replacing t...

Based on quantum graph theory we establish that the ray-splitting trace formula proposed by Couchman et al. (Phys. Rev. A 46, 6193 (1992)) is exact for a class of one-dimensional ray-splitting systems. Important applications in combinatorics are suggested. PACS: 05.45.+b,03.65.Sq,72.15.Rn

Let M1 and M2 be compact, orientable 3-manifolds, and M the manifold obtained by gluing some component F of ∂M1 to some component of ∂M2 by a homeomorphism φ. We show that when φ is “sufficiently complicated” then (1) the amalgamation of low genus, unstabilized, boundary-unstabilized Heegaard splittings of Mi is an unstabilized splitting of M , (2) every low genus, unstabilized Heegaard splitti...

We prove better lower bounds on additive spanners and emulators, which are lossy compression schemes for undirected graphs, as well as lower bounds on shortcut sets, which reduce the diameter of directed graphs. We show that any O(n)-size shortcut set cannot bring the diameter below Ω(n1/6), and that any O(m)-size shortcut set cannot bring it below Ω(n1/11). These improve Hesse’s [15] lower bou...

We prove that all planar Laman graphs (i.e. minimally generically rigid graphs with a non-crossing planar embedding) can be generated from a single edge by a sequence of vertex splits. It has been shown recently [6,12] that a graph has a pointed pseudo-triangular embedding if and only if it is a planar Laman graph. Due to this connection, our result gives a new tool for attacking problems in th...

Data Two data sets were tested : the first one consisted of older normal mode splitting measurements (1, 2) (with 15 degree two and 7 degree four structure coefficients), and the second one consisted of the most recent splitting measurements that followed the great Bolivia and Kuril Islands earthquakes in 1994 (3–5) (with 22 degree two and 16 degree four data). The choice of data set one was ba...

We say that a graph G is Class 0 if its pebbling number is exactly equal to its number of vertices. For a positive integer d, let kðdÞ denote the least positive integer so that every graph G with diameter at most d and connectivity at least kðdÞ is Class 0. The existence of the function k was conjectured by Clarke, Hochberg and Hurlbert, who showed that if the function k exists, then it must sa...

Suppose m and t are integers such that 0 < t ≤ m. An (m, t)-splitting system is a pair (X,B), where |X| = m and B is a set of subsets of X, called blocks, such that for every Y ⊆ X and |Y | = t, there exists a block B ∈ B such that |B ∩Y | = bt/2c. An (m, t)splitting system is uniform if every block has size bm/2c. We present new construction methods of uniform splitting systems for t = 3 that ...