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To support the fact-checking process, researchers have been recently developing automatic multimodal methods that gather and analyze external information, evidence, related to the image-text pairs under examination. However, prior works assumed all external information collected from the web to be relevant. In this study, we introduce a \"Relevant Evidence Detection\" (RED) module to discern whether each piece of evidence is relevant, to support or refute the claim. Specifically, we develop the \"Relevant Evidence Detection Directed Transformer\" (RED-DOT) and explore multiple architectural variants (e.g., single or dual-stage) and mechanisms (e.g., \"guided attention\"). Extensive ablation and comparative experiments demonstrate that RED-DOT achieves significant improvements over the state-of-the-art (SotA) on the VERITE benchmark by up to 33.7%. Furthermore, our evidence re-ranking and element-wise modality fusion led to RED-DOT surpassing the SotA on NewsCLIPings+ by up to 3% without the need for numerous evidence or multiple backbone encoders. 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