Chartrand's Theorem

Chartrand's Theorem states that a connected graph is chordal if and only if every minimal cycle (the smallest loops that cannot be broken down further) of four or more vertices has a chord—a shortcut edge connecting two non-adjacent vertices within the cycle. In simple terms, in a chordal graph, all larger loops can be "broken" by a shortcut, preventing long, unbroken cycles. This property is crucial for efficient algorithms in areas like database theory and computer science, as it simplifies complex network structures by ensuring the absence of intricate cycles without shortcuts.