Seminars and Colloquia by Series

Optimal decompositions of quasi-line trigraphs

Series
Graph Theory Seminar
Time
Tuesday, October 25, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andrew King – Simon Fraser University
Chudnovsky and Seymour's structure theorem for quasi-line graphs has led to a multitude of recent results that exploit two structural operations: compositions of strips and thickenings. In this paper we prove that compositions of linear interval strips have a unique optimal strip decomposition in the absence of a specific degeneracy, and that every claw-free graph has a unique optimal antithickening, where our two definitions of optimal are chosen carefully to respect the structural foundation of the graph. Furthermore, we give algorithms to find the optimal strip decomposition in O(nm) time and find the optimal antithickening in O(m2) time. For the sake of both completeness and ease of proof, we prove stronger results in the more general setting of trigraphs. This gives a comprehensive "black box" for decomposing quasi-line graphs that is not only useful for future work but also improves the complexity of some previous algorithmic results. Joint work with Maria Chudnovsky.

Structure of crossing-critical graphs

Series
Graph Theory Seminar
Time
Friday, September 23, 2011 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Zdenek Dvorak – Charles University, Prague, Czech Republic
A graph G is k-crossing-critical if it cannot be drawn in plane with fewer than k crossings, but every proper subgraph of G has such a drawing. We aim to describe the structure of crossing-critical graphs. In this talk, we review some of their known properties and combine them to obtain new information regarding e.g. large faces in the optimal drawings of crossing-critical graphs. Based on joint work with P. Hlineny and L. Postle.

Points covered by many simplices

Series
Graph Theory Seminar
Time
Friday, September 16, 2011 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Daniel Kral – Charles University, Prague, Czech Republic
Boros and Furedi (for d=2) and Barany (for arbitrary d) proved that there exists a constant c_d>0 such that for every set P of n points in R^d in general position, there exists a point of R^d contained in at least c_d n!/(d+1)!(n-d-1)! (d+1)-simplices with vertices at the points of P. Gromov [Geom. Funct. Anal. 20 (2010), 416-526] improved the lower bound on c_d by topological means. Using methods from extremal combinatorics, we improve one of the quantities appearing in Gromov's approach and thereby provide a new stronger lower bound on c_d for arbitrary d. In particular, we improve the lower bound on c_3 from 0.06332 due to Matousek and Wagner to more than 0.07509 (the known upper bound on c_3 is 0.09375). Joint work with Lukas Mach and Jean-Sebastien Sereni.

Roman domination on 2-connected graphs

Series
Graph Theory Seminar
Time
Thursday, April 28, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chun-Hung Liu – Math, GT
A Roman dominating function of a graph G is a function f which maps V(G) to {0, 1, 2} such that whenever f(v)=0, there exists a vertex u adjacent to v such that f(u)=2. The weight of f is w(f) = \sum_{v \in V(G)} f(v). The Roman domination number \gamma_R(G) of G is the minimum weight of a Roman dominating function of G. Chambers, Kinnersley, Prince and West conjectured that \gamma_R(G) is at most the ceiling 2n/3 for any 2-connected graph G of n vertices. In this talk, we will give counter-examples to the conjecture, and proves that \gamma_R(G) is at most the maximum among the ceiling of 2n/3 and 23n/34 for any 2-connected graph G of n vertices. This is joint work with Gerard Jennhwa Chang.

Control of Multi-Robot Networks

Series
Graph Theory Seminar
Time
Thursday, April 21, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Magnus Egerstedt – ECE, GT
Arguably, the overarching scientific challenge facing the area of networked robot systems is that of going from local rules to global behaviors in a predefined and stable manner. In particular, issues stemming from the network topology imply that not only must the individual agents satisfy some performance constraints in terms of their geometry, but also in terms of the combinatorial description of the network. Moreover, a multi-agent robotic network is only useful inasmuch as the agents can be redeployed and reprogrammed with relative ease, and we address these two issues (local interactions and programmability) from a controllability point-of-view. In particular, the problem of driving a collection of mobile robots to a given target destination is studied, and necessary conditions are given for this to be possible, based on tools from algebraic graph theory. The main result will be a necessary condition for an interaction topology to be controllable given in terms of the network's external, equitable partitions.

On Steinberg's Conjecture: 3-coloring certain planar graphs

Series
Graph Theory Seminar
Time
Thursday, April 14, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Peter Whalen – Math, GT
Steinberg's Conjecture states that any planar graph without cycles of length four or five is three colorable. Borodin, Glebov, Montassier, and Raspaud showed that planar graphs without cycles of length four, five, or seven are three colorable and Borodin and Glebov showed that planar graphs without five cycles or triangles at distance at most two apart are three colorable. We prove a statement similar to both of these results: that any planar graph with no cycles of length four through six or cycles of length seven with incident triangles distance exactly two apart are three colorable. Special thanks to Robin Thomas for substantial contributions in the development of the proof.

A simple proof for the two disjoint odd cycles theorem

Series
Graph Theory Seminar
Time
Thursday, March 31, 2011 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kenta Ozeki – National Institute of Informatics, Japan
A characterization of graphs without an odd cycle is easy, of course,it is exactly bipartite. However, graphs without two vertex disjoint oddcycles are not so simple. Lovasz is the first to give a proof of the twodisjoint odd cycles theorem which characterizes internally 4-connectedgraphs without two vertex disjoint odd cycles. Note that a graph $G$ iscalled internally 4-connected if $G$ is 3-connected, and all 3-cutseparates only one vertex from the other.However, his proof heavily depends on the seminal result by Seymour fordecomposing regular matroids. In this talk, we give a new proof to thetheorem which only depends on the two paths theorem, which characterizesgraphs without two disjoint paths with specified ends (i.e., 2-linkedgraphs). In addition, our proof is simpler and shorter.This is a joint work with K. Kawarabayashi (National Institute ofInformatics).

3-Connected Minor Minimal Non-Projective Planar Graphs with an Internal 3-Separation

Series
Graph Theory Seminar
Time
Thursday, March 17, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Arash Asadi – Math, GT
The property that a graph has an embedding in projective plane is closed under taking minors. So by the well known theorem of Robertson and Seymour, there exists a finite list of minor-minimal graphs, call it L, such that a given graph G is projective planar if and only if G does not contain any graph isomorphic to a member of L as a minor. Glover, Huneke and Wang found 35 graphs in L, and Archdeacon proved that those are all the members of L. In this talk we show a new strategy for finding the list L. Our approach is based on conditioning on the connectivity of a member of L. Assume G is a member of L. If G is not 3-connected then the structure of G is well understood. In the case that G is 3-connected, the problem breaks down into two main cases, either G has an internal separation of order three or G is internally 4-connected . In this talk we find the set of all 3-connected minor minimal non-projective planar graphs with an internal 3-separation. This is joint work with Luke Postle and Robin Thomas.

Lecture series on the disjoint paths algorithm

Series
Graph Theory Seminar
Time
Monday, March 14, 2011 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiiles 168
Speaker
Paul Wollan – School of Mathematics, Georgia Tech and University of Rome
This lecture will conclude the series. In a climactic finish the speaker will prove the Unique Linkage Theorem, thereby completing the proof of correctness of the Disjoint Paths Algorithm.

First-Fit is Linear on (r+s)-free Posets

Series
Graph Theory Seminar
Time
Thursday, March 10, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kevin Milans – University of South Carolina
First-Fit is an online algorithm that partitions the elements of a poset into chains. When presented with a new element x, First-Fit adds x to the first chain whose elements are all comparable to x. In 2004, Pemmaraju, Raman, and Varadarajan introduced the Column Construction Method to prove that when P is an interval order of width w, First-Fit partitions P into at most 10w chains. This bound was subsequently improved to 8w by Brightwell, Kierstead, and Trotter, and independently by Narayanaswamy and Babu. The poset r+s is the disjoint union of a chain of size r and a chain of size s. A poset is an interval order if and only if it does not contain 2+2 as an induced subposet. Bosek, Krawczyk, and Szczypka proved that if P is an (r+r)-free poset of width w, then First-Fit partitions P into at most 3rw^2 chains and asked whether the bound can be improved from O(w^2) to O(w). We answer this question in the affirmative. By generalizing the Column Construction Method, we show that if P is an (r+s)-free poset of width w, then First-Fit partitions P into at most 8(r-1)(s-1)w chains. This is joint work with Gwena\"el Joret.

Pages