The vertex separator problem: a polyhedral investigation

The vertex separator (VS) problem in a graph G=(V,E) asks for a partition of V into nonempty subsets A, B, C such that there is no edge between A and B, and |C| is minimized subject to a bound on max{|A|,|B|}. We give a mixed integer programming formulation of the problem and investigate the vertex separator polytope (VSP), the convex hull of incidence vectors of vertex separators. Necessary and sufficient conditions are given for the VSP to be full dimensional. Central to our investigation is the relationship between separators and dominators. Several classes of valid inequalities are investigated, along with the conditions under which they are facet defining for the VSP. Some of our proofs combine in new ways projection with lifting.In a companion paper we develop a branch-and-cut algorithm for the (VS) problem based on the inequalities discussed here, and report on computational experience with a wide variety of (VS) problems drawn from the literature and inspired by various applications.Research supported by the National Science Foundation through grant #DMI-0098427 and by the Office of Naval Research through contract N00014-97-1-0196Research supported by the Brazilian agencies FAPESP (grant 01/14205–6), CAPES (grant BEX 04444/02–2) and CNPq (grants 302588/02–7 and Pronex 664107/97–4)     

Embedding nearly-spanning bounded degree trees

We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1 − ε)n vertices, in terms of the expansion properties of G. As a result we show that for fixed d ≥ 2 and 0 < ε < 1, there exists a constant c = c(d, ε) such that a random graph G(n, c/n) contains almost surely a copy of every tree T on (1 − ε)n vertices with maximum degree at most d. We also prove that if an (n, D, λ)-graph G (i.e., a D-regular graph on n vertices all of whose eigenvalues, except the first one, are at most λ in their absolute values) has large enough spectral gap D/λ as a function of d and ε, then G has a copy of every tree T as above. Research supported in part by a USA-Israeli BSF grant, by NSF grant CCR-0324906, by a Wolfensohn fund and by the State of New Jersey. Research supported in part by USA-Israel BSF Grant 2002-133, and by grants 64/01 and 526/05 from the Israel Science Foundation. Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, and by an Alfred P. Sloan fellowship.     

Finite modules over non-semisimple group rings

LetG be a fixed graph and letX _G be the number of copies ofG contained in the random graphG(n, p). We prove exponential bounds on the upper tail ofX _G which are best possible up to a logarithmic factor in the exponent. Our argument relies on an extension of Alon’s result about the maximum number of copies ofG in a graph with a given number of edges. Similar bounds are proved for the random graphG(n, M) too. Research of the second author supported by KBN grant 2 P03A 027 22. Research of the third author supported by KBN grant 2 P03A 15 23.     

Recognizing Berge Graphs

A graph is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. In this paper we give an algorithm to test if a graph G is Berge, with running time O(|V (G)|⁹). This is independent of the recent proof of the strong perfect graph conjecture.* Currently this author is a Clay Mathematics Institute Research Fellow.** Supported by NSF grant DMI-0352885 and ONR grant N00014-97-1-0196. Supported by ONR grant N00014-01-1-0608, and NSF grant DMS-0070912. Supported by EPSRC grant GR/R35629/01.     

Partial geometries and the triality quadric

Up to now there are eight partial geometries pg(7,8,4) known. Their point graphs as well as their block graphs are all related to the triality quadric Q⁺(7,2). We prove that some of these graphs are the point graph of (up to isomorphism) exactly one partial geometry. We investigate the relations among some of these eight partial geometries. Generalizing our results, we construct two new families of partial geometries pg(2^2n–1– 1, 2^2n–1, 2^2n–2).The second author is a Research Fellow supported by the Flemish Institute for the Promotion of Scientific and Technological Research in Industry (IWT), grant No. IWT/SB/971002.     

Amenability and weak amenability of the Fourier algebra

Let G be a locally compact group. We show that its Fourier algebra A(G) is amenable if and only if G has an abelian subgroup of finite index, and that its Fourier–Stieltjes algebra B(G) is amenable if and only if G has a compact, abelian subgroup of finite index. We then show that A(G) is weakly amenable if the component of the identity of G is abelian, and we prove some partial results towards the converse.Research supported by NSERC under grant no. 90749-00.Research supported by NSERC under grant no. 227043-00.     

Degree sums,k-factors and hamilton cycles in graphs

We prove (a generalization of) the following conjecture of R. Häggkvist: LetG be a 2-connected graph onn vertices where every pair of nonadjacent vertices has degree sum at leastn — k and assume thatG has ak-factor; thenG is hamiltonian. This result is a common generalization of well-known theorems of Ore and Jackson, respectively.The research for this paper was done while the second author visited Memphis State University, partly supported by a grant from the Netherlands Organization for Scientific Research (N.W.O.).     

Regular sets in residual classes

Sets regular modulo a fixed odd prime power are explicitly constructed under the condition that their cardinalities do not exceed an arbitrarily small positive power of the modulus.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 224–228, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 97-01-00721 and by the Professor B. Novak grant (Karlov University, Prague).     

Removable Edges in Longest Cycles of 4-Connected Graphs

Let G be a 4-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G–e; second, for all vertices x of degree 3 in G–e, delete x from G–e and then completely connect the 3 neighbors of x by a triangle. If multiple edges occur, we use single edges to replace them. The final resultant graph is denoted by Ge. If Ge is 4-connected, then e is called a removable edge of G. In this paper we obtain some results on removable edges in a longest cycle of a 4-connected graph G. We also show that for a 4-connected graph G of minimum degree at least 5 or girth at least 4, any edge of G is removable or contractible.Acknowledgment. The authors are greatly indebted to a referee for his valuable suggestions and comments, which are very helpful to improve the proof of our main result Lemma 3.3.Research supported by National Science Foundation of China AMS subject classification (2000): 05C40, 05C38, 05C75Final version received: March 10, 2004     

On two Hamilton cycle problems in random graphs

We study two problems related to the existence of Hamilton cycles in random graphs. The first question relates to the number of edge disjoint Hamilton cycles that the random graph G _n,p contains. δ(G)/2 is an upper bound and we show that if p ≤ (1 + o(1)) ln n/n then this upper bound is tight whp. The second question relates to how many edges can be adversarially removed from G _n,p without destroying Hamiltonicity. We show that if p ≥ K ln n/n then there exists a constant α > 0 such that whp G − H is Hamiltonian for all choices of H as an n-vertex graph with maximum degree Δ(H) ≤ αK ln n. Research supported in part by NSF grant CCR-0200945. Research supported in part by USA-Israel BSF Grant 2002-133 and by grant 526/05 from the Israel Science Foundation.     

Decomposing the complete graph into cycles of many lengths

For all oddv 3 the complete graph onvK _v vertices can be decomposed intov – 2 edge disjoint cycles whose lengths are 3, 3, 4, 5,...,v – 1. Also, for all oddv 7,K _v can be decomposed intov – 3 edge disjoint cycles whose lengths are 3, 4,...,v – 4,v – 2,v – 1,v. Research supported by Australian Research Council grant A49130102     

Polyhedral <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">2</Emphasis></Subscript>

 Using elementary graded automorphisms of polytopal algebras (essentially the coordinate rings of projective toric varieties) polyhedral versions of the group of elementary matrices and the Steinberg and Milnor groups are defined. They coincide with the usual K-theoretic groups in the special case when the polytope is a unit simplex and can be thought of as compact/polytopal substitutes for the tame automorphism groups of polynomial algebras. Relative to the classical case, many new aspects have to be taken into account. We describe these groups explicitly when the underlying polytope is 2-dimensional. Already this low-dimensional case provides interesting classes of groups. Received: 13 December 2001 / Revised version: 24 June 2002 The second author was supported by the Deutsche Forschungsgemeinschaft, INTAS grant 99-00817 and TMR grant ERB FMRX CT-97-0107 Mathematics Subject Classification (2000): 14L27, 14M25, 19C09, 52B20     

Algorithms for the matrix <Emphasis Type="Italic">p</Emphasis>th root

New theoretical results are presented about the principal matrix pth root. In particular, we show that the pth root is related to the matrix sign function and to the Wiener–Hopf factorization, and that it can be expressed as an integral over the unit circle. These results are used in the design and analysis of several new algorithms for the numerical computation of the pth root. We also analyze the convergence and numerical stability properties of Newtons method for the inverse pth root. Preliminary computational experiments are presented to compare the methods. AMS subject classification 15A24, 65H10, 65F30Numerical Analysis Report 454, Manchester Centre for Computational Mathematics, July 2004.Dario A. Bini: This work was supported by MIUR, grant number 2002014121.Nicholas J. Higham: This work was supported by Engineering and Physical Sciences Research Council grant GR/R22612 and by a Royal Society – Wolfson Research Merit Award.     

Decomposing Berge Graphs Containing No Proper Wheel, Long Prism Or Their Complements

In this paper we show that, if G is a Berge graph such that neither G nor its complement contains certain induced subgraphs, named proper wheels and long prisms, then either G is a basic perfect graph( a bipartite graph, a line graph of a bipartite graph or the complement of such graphs) or it has a skew partition that cannot occur in a minimally imperfect graph. This structural result implies that G is perfect. This work was supported in part by NSF grant DMI-0352885 and ONR grant N00014-03-1-0188.     

A Semidefinite Programming Based Polyhedral Cut and Price Approach for the Maxcut Problem

We investigate solution of the maximum cut problem using a polyhedral cut and price approach. The dual of the well-known SDP relaxation of maxcut is formulated as a semi-infinite linear programming problem, which is solved within an interior point cutting plane algorithm in a dual setting; this constitutes the pricing (column generation) phase of the algorithm. Cutting planes based on the polyhedral theory of the maxcut problem are then added to the primal problem in order to improve the SDP relaxation; this is the cutting phase of the algorithm. We provide computational results, and compare these results with a standard SDP cutting plane scheme. Research supported in part by NSF grant numbers CCR–9901822 and DMS–0317323. Work done as part of the first authors Ph.D. dissertation at RPI.     

Super Restricted Edge Connectivity of Regular Graphs

An edge cut of a connected graph is called restricted if it separates this graph into components each having order at least 2; a graph G is super restricted edge connected if G−S contains an isolated edge for every minimum restricted edge cut S of G. It is proved in this paper that k-regular connected graph G is super restricted edge connected if k > |V(G)|/2+1. The lower bound on k is exemplified to be sharp to some extent. With this observation, we determined the number of edge cuts of size at most 2k−2 of these graphs. Supported by NNSF of China (10271105); Ministry of Science and Technology of Fujian (2003J036); Education Ministry of Fujian (JA03147)     

Perfect couples of graphs

We generalize the concept of perfect graphs in terms of additivity of a functional called graph entropy. The latter is an information theoretic functional on a graphG with a probability distributionP on its vertex set. For any fixedP it is sub-additive with respect to graph union. The entropy of the complete graph equals the sum of those ofG and its complement G iffG is perfect. We generalize this recent result to characterize all the cases when the sub-additivity of graph entropy holds with equality.The research of the authors is partially supported by the Hungarian National Foundation for Scientific Research (OTKA), grant No. 1806 resp. No. 1812.     

Reconstruction of Sturm-Liouville differential operators with singularities inside the interval

The inverse spectral problem for Sturm-Liouville differential operators on a finite interval is studied for an arbitrary and finite number of regular singular points inside the interval. A uniqueness theorem is proved; necessary and sufficient conditions and a procedure for the solution of the inverse problem are obtained.Translated fromMatematicheskie Zametki, Vol. 64, No. 1, pp. 143–156, July, 1998.This research was supported by the Ministry of Education (KTsFE) under grant No. 96-1.7-4 and by the Russian Foundation for Basic Research under grant No. 97-01-00566.     

On the growth of sums of independent random variables

Some estimates of the growth of sums of independent random variables almost surely are established without any moment conditions. Bibliography: 6 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 158–164.This research was partially supported by the Russian Foundation for Basic Research, grant 02-01-00779, and by the Program Leading Scientific Schools, grant 00-15-96019.Translated by V. V. Petrov.