Schedule-at-a-Glance

11 PRIMA 2013 AbstractsEnumerating, counting, and determining the maximumnumber of various objects in graphs have long been establishedas important areas within graph theory and graphalgorithms. As the number of enumerated objects is veryoften exponential in the size of the input graph, enumerationalgorithms fall into two categories depending on theirrunning time: those whose running time is measured inthe size of the input, and those whose running time ismeasured in the size of the output. Based on this, weconcentrate on the following two types of algorithms.1. Exact exponential time algorithms. The design ofthese algorithms is mainly based on recursive branching.The running time is a function of the size of the inputgraph, and very often it also gives an upper bound on thenumber of enumerated objects any graph can have.2. Output polynomial algorithms. The running time ofthese algorithms is polynomial in the number of the enumeratedobjects that the input graph actually contains.Some of these algorithms have even better running timesin form of incremental polynomial or polynomial delay,depending on the time the algorithm spends between eachconsecutive object that is output.The methods for designing the two types of algorithmsare usually quite different. Common to both approachesis that efforts have traditionally mainly been concentratedon arbitray graphs, whereas graphs with particular structurehave largely been left unattended. In this talk welook at enumeration of objects in graphs with specialstructure. In particular, we focus on enumerating minimaldominating sets in various graph classes.Algorithms of type 1: The number of minimal dominatingsets that any graph on n vertices can have is knownto be at most 1.7159 n . This upper bound might not betight, since no examples of graphs with 1.5705 n or moreminimal dominating sets are known. For several classesof graphs, like chordal, split, and proper interval graphs,we substantially improve the upper bound on the numberof minimal dominating sets. At the same time, wegive algorithms for enumerating all minimal dominatingsets, where the running time of each algorithm is withina polynomial factor of the proved upper bound for thegraph class in question. In some cases, we provide examplesof graphs containing the maximum possible numberof minimal dominating sets for graphs in that class,thereby showing the corresponding upper bounds to betight.Algorithms of type 2: Enumeration of minimal dominatingsets in graphs has very recently been shown to beequivalent to enumeration of minimal transversals in hypergraphs.The question whether the minimal transversalsof a hypergraph can be enumerated in output polynomialtime is a fundamental and challenging question;it has been open for several decades and has triggeredextensive research. We show that all minimal dominatingsets of a line graph can be generated in incrementalpolynomial, and consequently output polynomial, time.We are able to improve the delay further on line graphsof bipartite graphs. Finally we show that our methodis also efficient on graphs of large girth, resulting in anincremental polynomial time algorithm to enumerate theminimal dominating sets of graphs of girth at least 7.The presentation is based on joint works with followingco-authors: Jean-Francois Couturier, Petr Golovach, Pimvan ’t Hof, Dieter Kratsch, and Yngve Villanger.Matrix partitionsPavol HellSimon Fraser University, Canadapavol@sfu.caI will survey recent work on vertex partitions of graphswith certain internal and certain external constraints.(These constraints are conveniently encoded in a matrix.)Many well studied examples, especially from the study ofperfect graphs, fit into this model. The results discussedwill include both algorithms and characterizations. Manyopen problem will be posed.The lexicographic methodJing HuangUniversity of Victoria, Canadajing@math.uvic.caLexicographic techniques have been identified as an importanttool in the design of graph algorithms. In thistalk I will give a short survey of how the techniques areused for solving graph problems such as recognitions, orientations,representations, and homomorphisms.On m-walk-regular graphs, a generalization ofdistance-regular graphsJack KoolenUniversity of Science and Technology of China, China andPohang University of Science and Technology, Koreakoolen@postech.ac.krIn this talk I will discuss m-walk-regular graphs. Thesegraphs were introduced by Fiol and Garriga in 1999 as ageneralization of distance-regular graphs. 0-Walk-regulargraphs were earlier introduced by Godsil and McKay aswalk-regular graphs. They showed that a graph G is walkregularif and only if the spectrum of G where I delete avertex x does not depend on the vertex x.Our main motivation to study m-walk-regular graphs is tounderstand the difference between m-walk-regular graphsand distance-regular graphs. We will show that manyresults on distance-regular graphs can be generalized to2-walk-regular graphs. We also give many examples of m-walk-regular graphs which are not distance-regular. Butwe also show that some results on distance-regular graphsare not true for 2-walk-regular graphs.Geometric representations of graphsJan KratochvilCharles University in Prague, Czech Republichonza@kam.mff.cuni.czGeometric representations of graphs are intensively studiedboth for their practical motivation and interestingcombinatorial properties. We will survey recent developmentin this area, including recent results on coloring geometricintersection graphs and computational complexityof extending partial geometric representations.The practice of graph isomorphismBrendan D. McKayAustralian National University, Australiabdm@cs.anu.edu.auWe are concerned with the practical aspects of computingthe automorphism groups of graphs, and determiningcanonical labellings of graphs. The speaker’s programnauty has been around since 1976, though it wasn’t calledthat until about 1983. Until a few years ago, there wasn’tvery much competition, but then came saucy, Bliss,conauto, and some other programs that could outperformnauty in many cases.Our aim in the talk is to describe our response to thechallenge. In particular, nauty is now bundled with ahighly innovative program called Traces. We contendthat the present edition of Traces is now the performancechampion. The talk will describe how these programswork and give a comparison between them.This is joint work with Adolfo Piperno (University ofRome).