We give a new, stronger proof that there are only finitely many k-vertex-critical (P5, gem)-free graphs for all k. Our proof further refines the structure of these graphs and allows for the implementation of a simple ...
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We give a new, stronger proof that there are only finitely many k-vertex-critical (P5, gem)-free graphs for all k. Our proof further refines the structure of these graphs and allows for the implementation of a simple exhaustive computer search to completely list all 6-and 7-vertex-critical (P5, gem)-free graphs. Our results imply the existence of polynomial-time certifying algorithms to decide the k-colourability of (P5, gem)-free graphs for all k where the certificate is either a k-colouring or a (k + 1)-vertex-critical induced subgraph. Our complete lists for k < 7 allow for the implementation of these algorithms for all k < 6.(c) 2023 Elsevier B.V. All rights reserved.
This paper introduces a notion of certified computation whereby an algorithm not only produces a result r for a given input x, but also proves that r is a correct result for x. This can greatly enhance the credibility...
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This paper introduces a notion of certified computation whereby an algorithm not only produces a result r for a given input x, but also proves that r is a correct result for x. This can greatly enhance the credibility of the result: if we trust the axioms and inference rules that are used in the proof, then we can be assured that r is correct. Typically, the reasoning used in a certified computation is much simpler than the computation itself. We present and analyze two examples of certifying algorithms. We have developed denotational proof languages (DPLs) as a uniform platform for certified computation. DPLs integrate computation and deduction seamlessly, offer strong soundness guarantees, and provide versatile mechanisms for constructing proofs and proof-search methods. We have used DPLs to implement numerous well-known algorithms as certifiers, ranging from sorting algorithms to compiler optimizations, the Hindley-Milner W algorithm, Prolog engines, and more.
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