Abstract. Modular decomposition is fundamental for many important problems in algorithmic graph theory including transitive orientation, the recognition of several classes of graphs, and certain combinatorial optimization problems. Accordingly, there has been a drive towards a practical, linear-time algorithm for the problem. This paper posits such an algorithm; we present a linear-time modular decomposition algorithm that proceeds in four straightforward steps. This is achieved by introducing the notion of factorizing permutations to an earlier recursive approach. The only data structure used is an ordered list of trees, and each of the four steps amounts to simple traversals of these trees. Previous algorithms were either exceedingly complicated or resorted to impractical data-structures.

Split decomposition of graphs was introduced by Cunningham (under the name join decomposition) as a generalization of the modular decomposition. This paper undertakes an investigation into the algorithmic properties of split decomposition. We do so in the context of graph-labelled trees (GLTs), a new combinatorial object designed to simplify its consideration. GLTs are used to derive an incremental characterization of split decomposition, with a simple combinatorial description, and to explore its properties with respect to Lexicographic Breadth-First Search (LBFS). Applying the incremental characterization to an LBFS ordering results in a split decomposition algorithm that runs in time O(n + m)α(n + m), where α is the inverse Ackermann function, whose value is smaller than 4 for any practical graph. Compared to Dahlhaus' linear time split decomposition algorithm [16], which does not rely on an incremental construction, our algorithm is just as fast in all but the asymptotic sense and full implementation details are given in this paper. Also, our algorithm extends to circle graph recognition, whereas no such extension is known for Dahlhaus' algorithm. The companion paper [25] uses our algorithm to derive the first sub-quadratic circle graph recognition algorithm.

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Circle graphs are the intersection graphs of chords in a circle. This paper presents the first sub-quadratic recognition algorithm for the class of circle graphs. Our algorithm is O(n + m) times the inverse Ackermann function, α(n + m), whose value is smaller than 4 for any practical graph. The algorithm is based on a new incremental Lexicographic Breadth-First Search characterization of circle graphs, and a new efficient data-structure for circle graphs, both developed in the paper. The algorithm is an extension of a Split Decomposition algorithm with the same running time developed by the authors in a companion paper.

We have adapted the methodology of Berry et al. (2012) for Intensity Modulated Radiotherapy (IMRT) and Volumetric Modulated Arc Therapy (VMAT) treatments at a fixed source to imager distance (SID) based on the manufacturer's through-air portal dose image prediction algorithm. In order to fix the SID a correction factor was introduced to account for the change in air gap between patient and imager. Commissioning data, collected with multiple field sizes, solid water thicknesses and air gaps, were acquired at 150 cm SID on the Varian aS1200 EPID. The method was verified using six IMRT and seven VMAT plans on up to three different phantoms. The method's sensitivity and accuracy were investigated by introducing errors. A global 3%/3 mm gamma was used to assess the differences between the predicted and measured portal dose images. The effect of a varying air gap on EPID signal was found to be significant - varying by up to 30% with field size, phantom thickness, and air gap. All IMRT plans passed the 3%/3 mm gamma criteria by more than 95% on the three phantoms. 23 of 24 arcs from the VMAT plans passed the 3%/3 mm gamma criteria by more than 95%. This method was found to be sensitive to a range of potential errors. The presented approach provides fast and accurate in-vivo EPID dosimetry for IMRT and VMAT treatments and can potentially replace many pre-treatment verifications.