The Structure of Graphs Whose Vertices of Degree at Least Three Are at a Distance of at Least Three

Key Points

• This study aims to characterize the structure of connected graphs where certain distance conditions apply to degree-three vertices.
• Defined a connected graph G with specific degree and distance constraints.
• Analyzed the relationships and configurations of vertices within G.
• Provided structural decomposition of graph G into a tree and a matching.
• Established that graph G can be decomposed into distinct structural components.
• Identified that G consists of a tree and a matching arrangement.