A simple linear-time algorithm for finding path-decompositions of small width

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.Comment: 9 page

Obstructions to within a few vertices or edges of acyclic

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: $k$-{\sc Feedback Vertex Set} and $k$-{\sc 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.Comment: 16 page

Proffessor Wilson's Address at the Princeton Sesquicentennial Celebration

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Scientific Literature

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Diopside Glass-Ceramics for Dental and Biomedical Applications.

A series of glass compositions with varying equimolar amounts of Na2O:Al2O3 were designed using Appen factors. High purity batch reagents were ball milled for 30 min and transferred to Pt/Rh crucible and melted in an electric furnace (EHF1700, Lenton, UK) at high temperatures then held for 90 min. The molten glass was poured into a graphite mould, annealed at 50oC below the glass transition temperature for 1h and the remaining glass was quenched into water. Glass frits were crushed and ball milled into powders with different particle sizes. Glass powders (PS<125μm) were heat-treated via two-step heat treatment cycles and air quenched. Experimental glasses and glass-ceramics were characterised using X-Ray Diffraction (XRD), Dilatometry, Scanning Electron Microscopy (SEM) and solid state Nuclear Magnetic Resonance (NMR)