What question did this study set out to answer?

This study aims to characterize the structure of connected graphs where certain distance conditions apply to degree-three vertices.

February 14, 2026Mathematics0 citationsOpen Access

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

SGSalman GhazalAmerican University of the Middle East SKSteve KaramAmerican University of the Middle East

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.

Abstract

Let G be a connected graph in which the distance between any two distinct vertices of degree of at least three is at least three. We find the structure of G. It turns out that G decomposes into a tree and a matching.

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