A graph G is k-vertex-critical if chi(G) = k but chi(G - v) 1 and all l > 0. Together with previous results, the only graphs H for which it is unknown if there are an infinite number of k-vertex-critical H-free gr...
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A graph G is k-vertex-critical if chi(G) = k but chi(G - v) < k for all v is an element of V (G) where chi(G) denotes the chromatic number of G. We show that there are only finitely many k-vertex-critical (P-3+lP(1))-free graphs for all k > 1 and all l > 0. Together with previous results, the only graphs H for which it is unknown if there are an infinite number of k-vertex-critical H-free graphs is H = (P-4 + lP(1)) for all l >= 1. We consider a restriction on the smallest open case, and show that there are only finitely many k-vertex-critical (gem, co-gem)-free graphs for all k, where gem= P-4 + P-1. To do this, we show the stronger result that every vertex-critical (gem, co-gem)-free graph is either complete or a clique expansion of C-5. This characterization allows us to give the complete list of all k-vertex-critical (gem, co-gem)-free graphs for all k <= 16.(c) 2023 Elsevier B.V. All rights reserved.
A certifying algorithm verifies the correctness of its output at runtime by producing a witness in addition to an input-output pair. If a witness predicate holds for the triple, the input-output pair is correct. A che...
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ISBN:
(纸本)9783319675312;9783319675305
A certifying algorithm verifies the correctness of its output at runtime by producing a witness in addition to an input-output pair. If a witness predicate holds for the triple, the input-output pair is correct. A checker algorithm decides the witness predicate. While certifying sequential algorithms are well-established, we consider distributed algorithms. In this paper, we investigate certifying distributed algorithms that verify their distributed output. While the witness predicate states a property in the network, for distributed checking, additional predicates are decided for each component. We illustrate the applicability by examples.
Over the last years, witness-based validation of verification results has become an established practice in software verification: An independent validator re-establishes verification results of a software verifier us...
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Over the last years, witness-based validation of verification results has become an established practice in software verification: An independent validator re-establishes verification results of a software verifier using verification witnesses, which are stored in a standardized exchange format. In addition to validation, such exchangable information about proofs and alarms found by a verifier can be shared across verification tools, and users can apply independent third-party tools to visualize and explore witnesses to help them comprehend the causes of bugs or the reasons why a given program is correct. To achieve the goal of making verification results more accessible to engineers, it is necessary to consider witnesses as first-class exchangeable objects, stored independently from the source code and checked independently from the verifier that produced them, respecting the important principle of separation of concerns. We present the conceptual principles of verification witnesses, give a description of how to use them, provide a technical specification of the exchange format for witnesses, and perform an extensive experimental study on the application of witness-based result validation, using the validators CPACHECKER, UAUTOMIZER, CPA-WITNESS2TEST, and FSHELL-WITNESS2TEST.
Let A, B, and C be three n×n matrices. We investigate the problem of verifying whether AB=C over the ring of integers and finding the correct product AB. Given that C is different from AB by at most k entries, we...
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