publications: - title: "On computing graph minor obstruction sets" author: - name: "Kevin Cattell" link: "https://researchr.org/alias/kevin-cattell" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Rodney G. Downey" link: "https://researchr.org/alias/rodney-g.-downey" - name: "Michael R. Fellows" link: "https://researchr.org/alias/michael-r.-fellows" - name: "Michael A. Langston" link: "https://researchr.org/alias/michael-a.-langston" year: "2000" doi: "http://dx.doi.org/10.1016/S0304-3975(97)00300-9" abstract: "The Graph Minor Theorem of Robertson and Seymour establishes nonconstructively that many natural graph properties are characterized by a finite set of forbidden substructures, the {\\it obstructions} for the property. We prove several general theorems regarding the computation of obstruction sets from other information about a family of graphs. The methods can be adapted to other partial orders on graphs, such as the immersion and topological orders. The algorithms are in some cases practical and have been implemented. Two new technical ideas are introduced. The first is a method of computing a stopping signal for search spaces of expanding pathwidth. This allows obstruction sets to be computed for the first time without the necessity of a prior bound on maximum obstruction width. The second idea is that of a {\\it second order congruence} for a graph property. This is an equivalence relation defined on finite sets of graphs that generalizes the recognizability congruence that is defined on single graphs. It is shown that the obstructions for a graph ideal can be effectively computed from an oracle for the canonical second-order congruence for the ideal and a membership oracle for the ideal. It is shown that the obstruction set for a union ${\\cal F}= {\\cal F}_{1} \\cup {\\cal F}_{2}$ of minor ideals can be computed from the obstruction sets for ${\\cal F}_{1}$ and ${\\cal F}_{2}$ if there is at least one tree that does not belong to the intersection of ${\\cal F}_{1}$ and ${\\cal F}_{2}$. As a corollary, it is shown that the set of intertwines of an arbitrary graph and a tree are effectively computable. " links: doi: "http://dx.doi.org/10.1016/S0304-3975(97)00300-9" tags: - " obstruction sets" - "forbidden minors" - "graph minors" - "graph-rewriting" - "rewriting" - "search" researchr: "https://researchr.org/publication/CattellDDFL00" cites: 0 citedby: 0 journal: "TCS" volume: "233" number: "1-2" pages: "107-127" kind: "article" key: "CattellDDFL00" - title: "Construction of Time Relaxed Minimal Broadcast Networks" author: - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Jose A. Ventura" link: "https://researchr.org/alias/jose-a.-ventura" - name: "Mark C. Wilson" link: "https://researchr.org/alias/mark-c.-wilson" - name: "Golbon Zakeri" link: "https://researchr.org/alias/golbon-zakeri" year: "1999" abstract: "In broadcasting, or one-to-all communication, a message originally held in one node of the network must be transmitted to all the other nodes. A minimal broadcast network is a communication network that can transmit a message originated at any node to all other nodes of the network in minimum time. In this paper, we present a compound method to construct sparse, time-relaxed, minimal broadcast networks ($t$-mbn), in which broadcasting can be accomplished in slightly more than the minimum time. The proposed method generates a new network by connecting a subset of nodes from several copies of a $t_1$-mbn using the structure of another $t_2$-mbn. The objective is to construct a network as sparse as possible satisfying the desired broadcasting time constraint. Computational results illustrate the effectiveness of the proposed method." tags: - " time relaxed" - "constraints" - "C++" - "broadcast networks" researchr: "https://researchr.org/publication/DinneenVWZ99" cites: 0 citedby: 0 journal: "ppl" volume: "9" number: "1" pages: "53-68" kind: "article" key: "DinneenVWZ99" - title: "A fast natural algorithm for searching" author: - name: "Joshua J. Arulanandham" link: "https://researchr.org/alias/joshua-j.-arulanandham" - name: "Cristian Calude" link: "http://www.cs.auckland.ac.nz/~cristian" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" year: "2004" doi: "http://dx.doi.org/10.1016/j.tcs.2004.03.040" abstract: "In this note we present two natural algorithms---one for sorting and another for searching a sorted list of items. Both algorithms work in $O(\\sqrt N)$ time, $N$ being the size of the list. A combination of these algorithms can search an unsorted list in $O(\\sqrt N)$ time, an impossibility for classical algorithms. The same complexity is achieved by Grover's quantum search algorithm; in contrast to Grover's algorithm which is probabilistic, our method is guaranteed correct. Two applications will conclude this note. " links: doi: "http://dx.doi.org/10.1016/j.tcs.2004.03.040" tags: - "beads" - "search" researchr: "https://researchr.org/publication/ArulanandhamCD04%3A0" cites: 0 citedby: 0 journal: "TCS" volume: "320" number: "1" pages: "3-13" kind: "article" key: "ArulanandhamCD04:0" - title: "Properties of vertex cover obstructions" author: - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Rongwei Lai" link: "https://researchr.org/alias/rongwei-lai" year: "2007" doi: "http://dx.doi.org/10.1016/j.disc.2007.01.003" abstract: "We study properties of the sets of minimal forbidden minors for the families of graphs having a vertex cover of size at most $k$. We denote this set by $\\Obs(\\VCfam{k})$ and call it the set of obstructions. Our main result is to give a tight vertex bound of $\\Obs(\\VCfam{k})$, and then confirm a conjecture made by Liu Xiong that there is a unique connected obstruction with maximum number of vertices for \\VCfam{k} and this graph is $C_{2k+1}$. We also find two iterative methods to generate graphs in $\\Obs(\\VCfam{(k+1)})$ from any graph in $\\Obs(\\VCfam{k})$. " links: doi: "http://dx.doi.org/10.1016/j.disc.2007.01.003" tags: - "vertexcover" - "graph-rewriting" - "C++" - "rewriting" researchr: "https://researchr.org/publication/DinneenL07" cites: 0 citedby: 0 journal: "DM" volume: "307" number: "21" pages: "2484-2500" kind: "article" key: "DinneenL07" - title: "Update Networks and Their Routing Strategies" author: - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Bakhadyr Khoussainov" link: "https://researchr.org/alias/bakhadyr-khoussainov" year: "2000" doi: "http://link.springer.de/link/service/series/0558/bibs/1928/19280127.htm" abstract: "We introduce the notion of update networks to model communication networks with infinite duration. In our formalization we use bipartite finite graphs and game-theoretic terminology as an underlying structure. For these networks we exhibit a simple routing procedure to update information throughout the nodes of the network. We also introduce an hierarchy for the class of all update networks and discuss the complexity of some natural problems. " links: doi: "http://link.springer.de/link/service/series/0558/bibs/1928/19280127.htm" tags: - " finite graphs" - "networks" - "meta-model" - "update games" - "strategies" - "graph-rewriting" - "information models" - "Meta-Environment" - "rewriting" - "routing" - " infinite games" - "rewriting strategies" researchr: "https://researchr.org/publication/DinneenK00" cites: 0 citedby: 0 pages: "127-136" booktitle: "criwg" kind: "inproceedings" key: "DinneenK00" - title: "A Characterization of Graphs with Vertex Cover up to Five" author: - name: "Kevin Cattell" link: "https://researchr.org/alias/kevin-cattell" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" year: "1994" abstract: "For the family of graphs with fixed-size vertex cover $k$, we present all of the forbidden minors (obstructions), for $k$ up to five. We derive some results, including a practical finite-state recognition algorithm, needed to compute these obstructions." tags: - " vertex cover" - "forbidden minors" - "graph minors" - "graph-rewriting" - " obstructions" - "rewriting" researchr: "https://researchr.org/publication/CattellD94" cites: 0 citedby: 0 pages: "86-99" booktitle: "ordal" kind: "inproceedings" key: "CattellD94" - title: "A computational attack on the conjectures of Graffiti: New counterexamples and proofs" author: - name: "Tony L. Brewster" link: "https://researchr.org/alias/tony-l.-brewster" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Vance Faber" link: "https://researchr.org/alias/vance-faber" year: "1995" doi: "http://dx.doi.org/10.1016/0012-365X(94)00227-A" abstract: "Graffiti is a computer program that checks for relationships among certain graph invariants. It uses a database of graphs and has generated well over 700 conjectures. Having obtained a readily available computer tape of all the nonisomorphic graphs with 10 or fewer vertices, we have tested approximately 200 of the Graffiti conjectures and have found counterexamples for over 40 of them. For each conjecture that failed we display a counterexample. We also provide results that came from analyzing those conjectures which had a small number of counter- examples. Finally, we prove some results about four of the conjectures." links: doi: "http://dx.doi.org/10.1016/0012-365X(94)00227-A" tags: - "graffiti" - "testing" - "graph-rewriting" - "database" - "rewriting" - " automated math proving" researchr: "https://researchr.org/publication/BrewsterDF95" cites: 0 citedby: 0 journal: "DM" volume: "147" number: "1-3" pages: "35-55" kind: "article" key: "BrewsterDF95" - title: "A Simple Linear-Time Algorithm for Finding Path-Decompositions of Small Width" author: - name: "Kevin Cattell" link: "https://researchr.org/alias/kevin-cattell" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Michael R. Fellows" link: "https://researchr.org/alias/michael-r.-fellows" year: "1996" doi: "http://dx.doi.org/10.1016/0020-0190(95)00190-5" abstract: "We described a simple algorithm running in linear time for each fixed constant $k$, that either establishes that the pathwidth of a graph $G$ is greater than $k$, or finds a path-decomposition of $G$ of width at most $O(2^{k})$. This provides a simple proof of the result by Bodlaender that many families of graphs of bounded pathwidth can be recognized in linear time. " links: doi: "http://dx.doi.org/10.1016/0020-0190(95)00190-5" tags: - "graph-rewriting" - "algorithm" - "pathwidth" - "rewriting" researchr: "https://researchr.org/publication/CattellDF96" cites: 0 citedby: 0 journal: "ipl" volume: "57" number: "4" pages: "197-203" kind: "article" key: "CattellDF96" - title: "An Optimal Family of Bounded-Degree Broadcast Networks" author: - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Nian Zhou" link: "https://researchr.org/alias/nian-zhou" year: "2003" doi: "http://www1.elsevier.com/gej-ng/31/29/23/134/23/show/Products/notes/index.htt#013" abstract: "Increasingly, the design of efficient computer networks and multi-processor configurations are considered important applications of computer science. There are some constraints in network design which are usually created by economic and physical limitations. One constraint is the bounded degree, which is the limited number of connections between one node to others. Another possible constraint is a bound on the time that a message can afford to take during a ``broadcast''. We present, for the first time, a set of largest-known directed networks satisfied specified bounds on node degree and broadcast time. We also presents a family of optimal ($\\Delta$, $\\Delta+1$) broadcast digraphs. That is, digraphs with a proven maximum number of nodes, having maximum degree $\\Delta$ and broadcast time at most $\\Delta+1$. " links: doi: "http://www1.elsevier.com/gej-ng/31/29/23/134/23/show/Products/notes/index.htt#013" tags: - "design science" - "constraints" - "optimal broadcast networks" - "e-science" - "design" researchr: "https://researchr.org/publication/DinneenZ03" cites: 0 citedby: 0 journal: "ENTCS" volume: "78" pages: "196-208" kind: "article" key: "DinneenZ03" - title: "Forbidden minors to graphs with small feedback sets" author: - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Kevin Cattell" link: "https://researchr.org/alias/kevin-cattell" - name: "Michael R. Fellows" link: "https://researchr.org/alias/michael-r.-fellows" year: "2001" doi: "http://dx.doi.org/10.1016/S0012-365X(00)00083-2" abstract: "Finite obstruction set characterizations for lower ideals in the minor order are guaranteed to exist by the Graph Minor Theorem. In this paper we characterize several families of graphs with small feedback sets, namely $k_1$-{\\sc Feedback Vertex Set}, $k_2$-{\\sc Feedback Edge Set} and VertexEdgeFamily{k_1}{k_2}, for small integer parameters $k_1$ and $k_2$. Our constructive methods can compute obstruction sets for any minor-closed family of graphs, provided the pathwidth (or treewidth) of the largest obstruction is known. " links: doi: "http://dx.doi.org/10.1016/S0012-365X(00)00083-2" tags: - "forbidden minors" - " feedback vertex" - "graph-rewriting" - " feedback edge" - "rewriting" researchr: "https://researchr.org/publication/DinneenCF01" cites: 0 citedby: 0 journal: "DM" volume: "230" number: "1-3" pages: "215-252" kind: "article" key: "DinneenCF01" - title: "Bead-Sort: A Natural Sorting Algorithm" author: - name: "Joshua J. Arulanandham" link: "https://researchr.org/alias/joshua-j.-arulanandham" - name: "Cristian Calude" link: "http://www.cs.auckland.ac.nz/~cristian" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" year: "2002" abstract: "Nature is not only a source of minerals and precious stones but is also a mine of algorithms. By observing and studying natural phenomena, computer algorithms can be extracted. In this note, a simple natural phenomenon is used to design a sorting algorithm for positive integers, called here Bead--Sort. The algorithm's run--time complexity ranges from $O(1)$ to $O(S)$ ($S$ is the sum of the input integers) depending on the user's perspective. Finally, three possible implementations are suggested. " tags: - "design complexity" - "source-to-source" - "beadsort" - "design" - "open-source" researchr: "https://researchr.org/publication/ArulanandhamCD02" cites: 0 citedby: 0 journal: "eatcs" volume: "76" pages: "153-161" kind: "article" key: "ArulanandhamCD02" - title: "Small Diameter Symmetric Networks from Linear Groups" author: - name: "Lowell Campbell" link: "https://researchr.org/alias/lowell-campbell" - name: "Gunnar E. Carlsson" link: "https://researchr.org/alias/gunnar-e.-carlsson" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Vance Faber" link: "https://researchr.org/alias/vance-faber" - name: "Michael R. Fellows" link: "https://researchr.org/alias/michael-r.-fellows" - name: "Michael A. Langston" link: "https://researchr.org/alias/michael-a.-langston" - name: "James W. Moore" link: "https://researchr.org/alias/james-w.-moore" - name: "Andrew P. Mullhaupt" link: "https://researchr.org/alias/andrew-p.-mullhaupt" - name: "Harlan B. Sexton" link: "https://researchr.org/alias/harlan-b.-sexton" year: "1992" abstract: "In this note is reported a collection of constructions of symmetric networks that provide the largest known values for the number of nodes that can be placed in a network of a given degree and diameter. Some of the constructions are in the range of current potential engineering significance. The constructions are Cayleey graphs of liinear groups obtained by experimental computation." tags: - " Cayley graphs" - "degree" - " diameter" - "graph-rewriting" - "rewriting" researchr: "https://researchr.org/publication/CampbellCDFFLMMS92" cites: 0 citedby: 0 journal: "TC" volume: "41" number: "2" pages: "218-220" kind: "article" key: "CampbellCDFFLMMS92" - title: "What is the Value of Taxicab(6)?" author: - name: "Cristian Calude" link: "http://www.cs.auckland.ac.nz/~cristian" - name: "Elena Calude" link: "https://researchr.org/alias/elena-calude" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" year: "2003" doi: "http://www.jucs.org/jucs_9_10/what_is_the_value" abstract: "For almost 350 years it was known that 1729 is the smallest integer which can be expressed as the sum of two positive cubes in two different ways. Motivated by a famous story involving Hardy and Ramanujan, a class of numbers called Taxi-cab Numbers has been defined: Taxicab(k,j,n) is the smallest number which can be expressed as the sum of j kth powers in n different ways. So, Taxicab(3,2,2) = 1729;Taxicab(4,2,2) = 635318657. Computing Taxicab Numbers is challenging and interesting, both from mathematical and programming points of view. The exact value of Taxicab(6) = Taxicab(3, 2, 6) is not known; however, recent results announced by Rathbun [R2002] show that Taxicab(6) is in the interval [1018,24153319581254312065344]. " links: doi: "http://www.jucs.org/jucs_9_10/what_is_the_value" tags: - "taxicab" - "programming" researchr: "https://researchr.org/publication/CaludeCD03" cites: 0 citedby: 0 journal: "jucs" volume: "9" number: "10" pages: "1196-1203" kind: "article" key: "CaludeCD03" - title: "Compound Constructions of Broadcast Networks" author: - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Jose A. Ventura" link: "https://researchr.org/alias/jose-a.-ventura" - name: "Mark C. Wilson" link: "https://researchr.org/alias/mark-c.-wilson" - name: "Golbon Zakeri" link: "https://researchr.org/alias/golbon-zakeri" year: "1999" doi: "http://dx.doi.org/10.1016/S0166-218X(99)00043-8" abstract: "Compound methods have been shown to be very effective in the construction of broadcast graphs. Compound methods generate a large broadcast graph by combining multiple copies of a broadcast graph $G$ using the structure of another broadcast graph $H$. Node deletion is also allowed in some of these methods. The subset of connecting nodes of $G$ has been defined as solid $h$-cover by Bermond, Fraigniaud and Peters, and center node set by Weng and Ventura. This article shows that the two concepts are equivalent. We also provide new properties for center node sets, including bounds on the minimum size of a center node set, show how to reduce the number of center nodes of a broadcast graph generated by a compound method, and propose an iterative compounding algorithm that generates the sparsest known broadcast graphs in many cases. " links: doi: "http://dx.doi.org/10.1016/S0166-218X(99)00043-8" tags: - "graph-rewriting" - "C++" - " compound graphs" - "rewriting" - "broadcast networks" researchr: "https://researchr.org/publication/DinneenVWZ99%3A0" cites: 0 citedby: 0 journal: "DAM" volume: "93" number: "2-3" pages: "205-232" kind: "article" key: "DinneenVWZ99:0" - title: "Update games and update networks" author: - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Bakhadyr Khoussainov" link: "https://researchr.org/alias/bakhadyr-khoussainov" year: "2003" doi: "http://dx.doi.org/10.1016/S1570-8667(03)00006-6" abstract: "In this paper we model infinite processes with finite configurations as infinite games over finite graphs. We investigate those games, called \\emph{update games}, in which each configuration occurs an infinite number of times during a two-person play. We also present an efficient polynomial-time algorithm (and partial characterization) for deciding if a graph is an \\emph{update network}. " links: doi: "http://dx.doi.org/10.1016/S1570-8667(03)00006-6" tags: - "meta-model" - "modeling" - "update games" - "graph-rewriting" - "Meta-Environment" - "rewriting" - "process modeling" researchr: "https://researchr.org/publication/DinneenK03" cites: 0 citedby: 0 journal: "jda" volume: "1" number: "1" pages: "53-65" kind: "article" key: "DinneenK03" - title: "Degree- and time-constrained broadcast networks" author: - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Geoffrey Pritchard" link: "https://researchr.org/alias/geoffrey-pritchard" - name: "Mark C. Wilson" link: "https://researchr.org/alias/mark-c.-wilson" year: "2002" doi: "http://dx.doi.org/10.1002/net.10018" abstract: "We consider the problem of constructing networks with as many nodes as possible, subject to upper bounds on the degree and broadcast time. This paper includes the results of an extensive empirical study of broadcasting in small regular graphs using a stochastic search algorithm to approximate the broadcast time. Significant improvements on known results are obtained for cubic broadcast networks. Some tests based on easily computable graph-theoretic properties are presented. " links: doi: "http://dx.doi.org/10.1002/net.10018" tags: - "empirical" - "rule-based" - "testing" - "graph-rewriting" - "C++" - "rewriting" - "search" - "broadcast networks" researchr: "https://researchr.org/publication/DinneenPW02" cites: 0 citedby: 0 journal: "networks" volume: "39" number: "3" pages: "121-129" kind: "article" key: "DinneenPW02" - title: "Obstructions to Within a Few Vertices or Edges of Acyclic" author: - name: "Kevin Cattell" link: "https://researchr.org/alias/kevin-cattell" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Michael R. Fellows" link: "https://researchr.org/alias/michael-r.-fellows" year: "1995" abstract: "Finite obstruction sets for lower ideals in the minor order are guaranteed to exist by the Graph Minor Theorem. It has been known for several years that, in principle, obstruction sets can be mechanically computed for most natural lower ideals. In this paper, we describe a general-purpose method for finding obstructions by using a bounded treewidth (or pathwidth) search. We illustrate this approach by characterizing certain families of cycle-cover graphs based on the two well-known problems: $k$-{Feedback Vertex Set} and $k$-{Feedback Edge Set}. Our search is based on a number of algorithmic strategies by which large constants can be mitigated, including a randomized strategy for obtaining proofs of minimality. " tags: - "rule-based" - "forbidden minors" - "graph-rewriting" - "rewriting" - "search" - " feedback edge set" - " feedback vertex set" - "systematic-approach" - "rewriting strategies" researchr: "https://researchr.org/publication/CattellDF95" cites: 0 citedby: 0 pages: "415-427" booktitle: "wads" kind: "inproceedings" key: "CattellDF95" - title: "Recent examples in the theory of partition graphs" author: - name: "D. W. DeTemple" link: "https://researchr.org/alias/d.-w.-detemple" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "J. M. Robertson" link: "https://researchr.org/alias/j.-m.-robertson" - name: "K. L. McAvaney" link: "https://researchr.org/alias/k.-l.-mcavaney" year: "1993" doi: "http://dx.doi.org/10.1016/0012-365X(93)90520-4" abstract: "A partition graph is an intersection graph for a collection of subsets of a universal set S with the property that every maximal independent set of vertices corresponds to a partition of S. Two questions which arose in the study of partition graphs are answered by recently discovered examples. An enumeration of the partition graphs on ten or fewer vertices is provided." links: doi: "http://dx.doi.org/10.1016/0012-365X(93)90520-4" tags: - "partition graphs" - "graph-rewriting" - "rewriting" - "partitioning" researchr: "https://researchr.org/publication/DeTempleDRM93" cites: 0 citedby: 0 journal: "DM" volume: "113" number: "1-3" pages: "255-258" kind: "article" key: "DeTempleDRM93" - title: "On Game-Theoretic Models of Networks" author: - name: "Hans Leo Bodlaender" link: "http://people.cs.uu.nl/hansb/" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Bakhadyr Khoussainov" link: "https://researchr.org/alias/bakhadyr-khoussainov" year: "2001" doi: "http://link.springer.de/link/service/series/0558/bibs/2223/22230550.htm" abstract: "In this paper, we study the complexity of deciding which player has a winning strategy in certain types of McNaughton games. These graph games can be used as models for computational problems and processes of infinite duration. We consider the cases (1) where the first player wins when vertices in a specified set are visited infinitely often and vertices in another specified set are visited finitely often, (2) where the first player wins when exactly those vertices in one of a number of specified disjoint sets are visited infinitely often, and (3) a generalization of these first two cases. We give polynomial time algorithms to determine which player has a winning strategy in each of the games considered. " links: doi: "http://link.springer.de/link/service/series/0558/bibs/2223/22230550.htm" tags: - " finite graphs" - "case study" - "meta-model" - "modeling" - "graph-rewriting" - "game networks" - "Meta-Environment" - "rewriting" - " infinite games" - "process modeling" - "rewriting strategies" researchr: "https://researchr.org/publication/BodlaenderDK01" cites: 0 citedby: 0 pages: "550-561" booktitle: "isaac" kind: "inproceedings" key: "BodlaenderDK01" - title: "Bounded Combinatorial Width and Forbidden Substructures" author: - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" year: "1995" month: "dec" abstract: "A substantial part of graph theory's history deals with the study and classification of sets of graphs that share common properties. One predominant trend is to characterized graph families by sets of minimal forbidden graphs (within some partial ordering of the graphs). For example, the famous Kuratowski Theorem classifies the planar graph family (i.e., those graphs that can be drawn on the plane without edge crossings) by two forbidden graphs (in the topological partial order). Most, if not all, of the current approaches for finding these forbidden substructure characterizations use extensive and specialized case analysis. Thus, until now, for a fixed graph family, this type of mathematical theorem proving often required months or even years of human effort. The main focus of this dissertation is to develop a practical theory for automating (with distributed computer programming) this classic part of graph theory. The recently celebrated Robertson--Seymour Graph Minor Theorem establishes the fact that many natural graph families are characterizable by a {\\em finite} set of graphs. In particular, if a graph family is closed under the three basic minor operations (i.e., isolated vertex deletions, edge deletions, and edge contractions) then there exists (by a nonconstructive argument) a finite set of forbidden graphs. Two minor-order closed examples are the well-known {$k$-Vertex Cover} and {$k$-Feedback Vertex Set} graph families. In this dissertation, we characterize, for the first time, these parameterized families, among others, for small $k$. A key ingredient in our forbidden graph computations is our reliance on the concept of bounded combinatorial width (e.g., pathwidth and treewidth). Using an algebraic graph enumeration scheme for all graphs of a fixed combinatorial width, we have implemented a terminating algorithm that will, find all minor-order forbidden graphs bounded by that width. For a targeted graph family, this algorithm requires a mathematical description given in one of many acceptable forms (or combinations thereof), such as a finite-state congruence or a set of automaton-generating tests. Our main assumption is that an upper bound on the pathwidth (or treewidth) of the largest forbidden graph of a particular graph family is more readily available than its order (or size). And, more importantly, a width bound is more sharp, yielding a smaller search space. A byproduct of our bounded width approach is that we give practical linear-time membership algorithms in the form of dynamic programs (over parsed graph structures of bounded width) for several graph families (e.g., {$k$-Maximum Path Length} and {$k$-Outer Planar}). " tags: - "parsing algorithm" - "program analysis" - "classification" - "case study" - "forbidden minors" - "testing" - "analysis" - "planar graph" - "graph-rewriting" - " obstructions" - " algebra" - "programming" - "type theory" - "history" - "rewriting" - "search" - "parsing" - "systematic-approach" researchr: "https://researchr.org/publication/DiPhd95" cites: 0 citedby: 0 school: "Dept. of Computer Science, University of Victoria" address: "P.O. Box 3055, Victoria, B.C., Canada~~V8W~3P6" advisor: - name: "Michael R. Fellows" link: "http://www.mrfellows.net/" kind: "phdthesis" key: "DiPhd95" - title: "Logic in Computer Science" author: - name: "Douglas S. Bridges" link: "https://researchr.org/alias/douglas-s.-bridges" - name: "Cristian Calude" link: "http://www.cs.auckland.ac.nz/~cristian" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Bakhadyr Khoussainov" link: "https://researchr.org/alias/bakhadyr-khoussainov" year: "1997" doi: "http://www.jucs.org/jucs_3_11/logic_in_computer_science" abstract: "On 8 August 1997, prompted by the visit of Hajime Ishihara from JAIST (the Japan Advanced Institute of Science & Technology, Hokoriku), the Centre for Discrete Mathematics & Theoretical Computer Science held a one-day workshop on Logic in Computer Science at the University of Auckland, New Zealand. This workshop brought together a number of mathematicians and computer scientists interested in computability, complexity, constructive mathematics, and other aspects of theoretical computer science. The proceedings of the workshop, having been subjected to the normal process of refereeing, are now collected together in this special issue of J. UCS, where the reader will find a wide variety of papers, dealing with subjects ranging from constructive aspects of the Dirichlet Problem to pure recursion theory and from partial orders to automata theory. " links: doi: "http://www.jucs.org/jucs_3_11/logic_in_computer_science" tags: - "automata theory" - "e-science" - "logic" researchr: "https://researchr.org/publication/BridgesCDK97" cites: 0 citedby: 0 journal: "jucs" volume: "3" number: "11" pages: "1133" kind: "article" key: "BridgesCDK97" - title: "Relaxed Update and Partition Network Games" author: - name: "Hans Leo Bodlaender" link: "http://people.cs.uu.nl/hansb/" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Bakhadyr Khoussainov" link: "https://researchr.org/alias/bakhadyr-khoussainov" year: "2002" abstract: "In this paper, we study the complexity of deciding which player has a winning strategy in certain types of McNaughton games. These graph games can be used as models for computational problems and processes of infinite duration. We consider the cases (1) where the first player wins when vertices in a specified set are visited infinitely often and vertices in another specified set are visited finitely often, (2) where the first player wins when exactly those vertices in one of a number of specified disjoint sets are visited infinitely often, and (3) a generalization of these first two cases. We give polynomial time algorithms to determine which player has a winning strategy in each of the games considered. " tags: - "case study" - "meta-model" - "relaxed update network games" - "graph-rewriting" - "Meta-Environment" - "rewriting" - "partitioning" - "process modeling" - "rewriting strategies" researchr: "https://researchr.org/publication/BodlaenderDK02" cites: 0 citedby: 0 journal: "FUIN" volume: "49" number: "4" pages: "301-312" kind: "article" key: "BodlaenderDK02" - title: "Algebraic Constructions of Efficient Broadcast Networks" author: - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Michael R. Fellows" link: "https://researchr.org/alias/michael-r.-fellows" - name: "Vance Faber" link: "https://researchr.org/alias/vance-faber" year: "1991" abstract: "Cayley graph techniques are introduced for the problem of constructing networks having the maximum possible number of nodes, among networks that satisfy prescribed bounds on the parameters maximum node degree and broadcast diameter. The broadcast diameter of a network is the maximum time required for a message originating at a node of the network to be relayed to all other nodes, under the restriction that in a single time step any node can communicate with only one neighboring node. For many parameter values these algebraic methods yield the largest known constructions, improving on previous graph-theoretic approaches. It has previously been shown that hypercubes are optimal for degree k and broadcast diameter k. A construction employing dihedral groups is shown to be optimal for degree k and broadcast diameter k + 1. " tags: - " network design" - " Cayley graphs" - "graph-rewriting" - " algebra" - "rewriting" - "systematic-approach" researchr: "https://researchr.org/publication/DinneenFF91" cites: 0 citedby: 0 pages: "152-158" booktitle: "AAECC" kind: "inproceedings" key: "DinneenFF91" - title: "A Characterization of Graphs with Vertex Cover Six" author: - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Liu Xiong" link: "https://researchr.org/alias/liu-xiong" year: "2000" doi: "http://link.springer.de/link/service/series/0558/bibs/1858/18580180.htm" abstract: "The complete list of forbidden minors (obstructions) for the family of graphs with vertex cover 6 have been found. This paper shows how one can limit the search space of graphs and how to simplify the process for deciding whether a graph is an obstruction for \\VC{k}. The upper bounds $2k+1$ ($2k+2$) on the maximum number of vertices for connected (disconnected) obstructions are shown to be sharp for all $k > 0$. " links: doi: "http://link.springer.de/link/service/series/0558/bibs/1858/18580180.htm" tags: - "completeness" - " vertex cover" - "forbidden minors" - "graph-rewriting" - "rewriting" - "search" researchr: "https://researchr.org/publication/DinneenX00" cites: 0 citedby: 0 pages: "180-192" booktitle: "cocoon" kind: "inproceedings" key: "DinneenX00" - title: "Algebraic Methods for Efficient Network Constructions" author: - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" year: "1991" month: "oct" abstract: "The design of computer networks and parallel processor configurations is a topic of increasing importance. Network designs which efficiently support communications between nodes are crucial for many applications. Cost and physical limitations generally prevent the nodes in a network from having more than a fixed number of hardware connections to other nodes (that is, the nodes must have bounded degree). This fundamental constraint makes the design problem nontrivial. The topic of this thesis is an explanation of ways in which group theory can be used to design bounded-degree communication-efficient networks. Our methods have yielded a number of network designs that are the largest known for networks satisfying specified bounds on node degree and either diameter or broadcast time, for values of these parameters that are in the range of potential engineering significance. " tags: - "process algebra" - "algebraic network design" - "constraints" - " algebra" - "design" researchr: "https://researchr.org/publication/DiMasters91" cites: 0 citedby: 0 school: "Dept. of Computer Science, University of Victoria" type: "Master's Thesis" address: "P.O. Box 3055, Victoria, B.C., Canada~~V8W~3P6" advisor: - name: "Michael R. Fellows" link: "http://www.mrfellows.net/" kind: "mastersthesis" key: "DiMasters91" - title: "Exact Approximations of omega Numbers" author: - name: "Cristian S. Calude" link: "http://www.cs.auckland.ac.nz/~cristian" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" year: "2007" doi: "http://dx.doi.org/10.1142/S0218127407018130" abstract: "A Chaitin Omega number is the halting probability of a universal Chaitin (self-delimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly non-computable. The aim of this paper is to describe a procedure, which combines Java programming and mathematical proofs, for computing the exact values of the first 64 bits of a Chaitin Omega: 0000001000000100000110001000011010001111110010111011101000010000. Full description of programs and proofs will be given elsewhere. " links: doi: "http://dx.doi.org/10.1142/S0218127407018130" tags: - "Java" - "omega" - "programming" researchr: "https://researchr.org/publication/CaludeD07" cites: 0 citedby: 0 journal: "IJBC" volume: "17" number: "6" pages: "1937-1954" kind: "article" key: "CaludeD07" - title: "Balance Machines: Computing = Balancing" author: - name: "Joshua J. Arulanandham" link: "https://researchr.org/alias/joshua-j.-arulanandham" - name: "Cristian Calude" link: "http://www.cs.auckland.ac.nz/~cristian" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" year: "2004" doi: "http://springerlink.metapress.com/openurl.asp?genre=article&issn=0302-9743&volume=2950&spage=36" abstract: "We propose a natural computational model called a balance machine. The computational model consists of components that resemble ordinary physical balances, each with an intrinsic property to automatically balance the weights on their left, right pans. If we start with certain fixed weights (representing inputs) on some of the pans, then the balance--like components would vigorously try to balance themselves by filling the rest of the pans with suitable weights (representing the outputs). This balancing act can be viewed as a computation. We will show that the model allows us to construct those primitive (hardware) components that serve as the building blocks of a general purpose (universal) digital computer: logic gates, memory cells (flip-flops) and transmission lines. One of the key features of the balance~machine is its ``bidirectional\" operation: both a function and its (partial) inverse can be computed spontaneously using the same machine. Some practical applications of the model are discussed." links: doi: "http://springerlink.metapress.com/openurl.asp?genre=article&issn=0302-9743&volume=2950&spage=36" tags: - "meta-model" - "modeling" - "balance machines" - "logic" - "Meta-Environment" researchr: "https://researchr.org/publication/ArulanandhamCD04" cites: 0 citedby: 0 pages: "36-48" booktitle: "BIRTHDAY" kind: "inproceedings" key: "ArulanandhamCD04" - title: "Too Many Minor Order Obstructions" author: - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" year: "1997" doi: "http://www.jucs.org/jucs_3_11/too_many_minor_order" abstract: "We study the growth rate on the number obstructions (forbidden minors) for families of graphs that are based on parameterized graph problems. Our main result shows that if the defining graph problem is \\NP-complete then the growth rate on the number of obstructions must be super-polynomial or else the polynomial-time hierarchy must collapse to $\\Sigma^P_3$. We illustrate the rapid growth rate of parameterized lower ideals by computing (and counting) the obstructions for the graph families with independence plus size at most $k$, $k \\leq 12$. " links: doi: "http://www.jucs.org/jucs_3_11/too_many_minor_order" tags: - "rule-based" - "completeness" - "forbidden minors" - "graph-rewriting" - " obstructions" - "rewriting" - "minor order" researchr: "https://researchr.org/publication/Dinneen97" cites: 0 citedby: 0 journal: "jucs" volume: "3" number: "11" pages: "1199-1206" kind: "article" key: "Dinneen97" - title: "Unconventional Models of Computation, Third International Conference, UMC 2002, Kobe, Japan, October 15-19, 2002, Proceedings" year: "2002" researchr: "https://researchr.org/publication/umc%3A2002" cites: 0 citedby: 0 booktitle: "Unconventional Models of Computation, Third International Conference, UMC 2002, Kobe, Japan, October 15-19, 2002, Proceedings" conference: "umc" editor: - name: "Cristian Calude" link: "http://www.cs.auckland.ac.nz/~cristian" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Ferdinand Peper" link: "https://researchr.org/alias/ferdinand-peper" volume: "2509" series: "Lecture Notes in Computer Science" publisher: "Springer" isbn: "3-540-44311-8" kind: "proceedings" key: "umc:2002" - title: "Discrete Mathematics and Theoretical Computer Science, 4th International Conference, DMTCS 2003, Dijon, France, July 7-12, 2003. Proceedings" year: "2003" tags: - "e-science" researchr: "https://researchr.org/publication/dmtcs%3A2003" cites: 0 citedby: 0 booktitle: "Discrete Mathematics and Theoretical Computer Science, 4th International Conference, DMTCS 2003, Dijon, France, July 7-12, 2003. Proceedings" conference: "dmtcs" editor: - name: "Cristian Calude" link: "http://www.cs.auckland.ac.nz/~cristian" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Vincent Vajnovszki" link: "https://researchr.org/alias/vincent-vajnovszki" volume: "2731" series: "Lecture Notes in Computer Science" publisher: "Springer" isbn: "3-540-40505-4" kind: "proceedings" key: "dmtcs:2003" - title: "Developments in Language Theory, 8th International Conference, DLT 2004, Auckland, New Zealand, December 13-17, 2004, Proceedings" year: "2004" researchr: "https://researchr.org/publication/dlt%3A2004" cites: 0 citedby: 0 booktitle: "Developments in Language Theory, 8th International Conference, DLT 2004, Auckland, New Zealand, December 13-17, 2004, Proceedings" conference: "dlt" editor: - name: "Cristian Calude" link: "http://www.cs.auckland.ac.nz/~cristian" - name: "Elena Calude" link: "https://researchr.org/alias/elena-calude" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" volume: "3340" series: "Lecture Notes in Computer Science" publisher: "Springer" isbn: "3-540-24014-4" kind: "proceedings" key: "dlt:2004" - title: "Unconventional Computation, 5th International Conference, UC 2006, York, UK, September 4-8, 2006, Proceedings" year: "2006" researchr: "https://researchr.org/publication/uc%3A2006" cites: 0 citedby: 0 booktitle: "Unconventional Computation, 5th International Conference, UC 2006, York, UK, September 4-8, 2006, Proceedings" editor: - name: "Cristian S. Calude" link: "http://www.cs.auckland.ac.nz/~cristian" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Gheorghe Paun" link: "https://researchr.org/alias/gheorghe-paun" - name: "Grzegorz Rozenberg" link: "https://researchr.org/alias/grzegorz-rozenberg" - name: "Susan Stepney" link: "https://researchr.org/alias/susan-stepney" volume: "4135" series: "Lecture Notes in Computer Science" publisher: "Springer" isbn: "3-540-38593-2" kind: "proceedings" key: "uc:2006" - title: "Unconventional Computation, 6th International Conference, UC 2007, Kingston, Canada, August 13-17, 2007, Proceedings" year: "2007" researchr: "https://researchr.org/publication/uc%3A2007" cites: 0 citedby: 0 booktitle: "Unconventional Computation, 6th International Conference, UC 2007, Kingston, Canada, August 13-17, 2007, Proceedings" editor: - name: "Selim G. Akl" link: "https://researchr.org/alias/selim-g.-akl" - name: "Cristian S. Calude" link: "http://www.cs.auckland.ac.nz/~cristian" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Grzegorz Rozenberg" link: "https://researchr.org/alias/grzegorz-rozenberg" - name: "Todd Wareham" link: "https://researchr.org/alias/todd-wareham" volume: "4618" series: "Lecture Notes in Computer Science" publisher: "Springer" isbn: "978-3-540-73553-3" kind: "proceedings" key: "uc:2007" - title: "Unconventional Computation, 4th International Conference, UC 2005, Sevilla, Spain, October 3-7, 2005, Proceedings" year: "2005" researchr: "https://researchr.org/publication/uc%3A2005" cites: 0 citedby: 0 booktitle: "Unconventional Computation, 4th International Conference, UC 2005, Sevilla, Spain, October 3-7, 2005, Proceedings" editor: - name: "Cristian Calude" link: "http://www.cs.auckland.ac.nz/~cristian" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" - name: "Gheorghe Paun" link: "https://researchr.org/alias/gheorghe-paun" - name: "Mario J. Pérez-Jiménez" link: "https://researchr.org/alias/mario-j.-p%C3%A9rez-jim%C3%A9nez" - name: "Grzegorz Rozenberg" link: "https://researchr.org/alias/grzegorz-rozenberg" volume: "3699" series: "Lecture Notes in Computer Science" publisher: "Springer" isbn: "3-540-29100-8" kind: "proceedings" key: "uc:2005" - title: "Unconventional Models of Computation, UMC 2K, Proceedings of the Second International Conference, Brussel, Belgium, 13-16 December 2000" year: "2001" researchr: "https://researchr.org/publication/umc%3A2000" cites: 0 citedby: 0 booktitle: "Unconventional Models of Computation, UMC 2K, Proceedings of the Second International Conference, Brussel, Belgium, 13-16 December 2000" conference: "umc" editor: - name: "Ioannis Antoniou" link: "https://researchr.org/alias/ioannis-antoniou" - name: "Cristian Calude" link: "http://www.cs.auckland.ac.nz/~cristian" - name: "Michael J. Dinneen" link: "http://www.cs.auckland.ac.nz/~mjd" publisher: "Springer" isbn: "1-85233-415-0" kind: "proceedings" key: "umc:2000"