Complex sentence: David Dumville, relying on the Chronicle alone, appears to accept that Eochaid was king, while Archie Duncan, arguing from the same source, flatly rejects the idea that Eochaid was king and attributes the supposed joint reign to Giric, and to Giric alone.
Split into simpler sentences: David Dumville, relying on the "Chronicle" alone, appears to accept that Eochaid was king. However, Archie Duncan, also relying on the "Chronicle", flatly rejects the idea that Eochaid was king and attributes the supposed joint reign to Giric, and to Giric alone.