This content establishes that NP-completeness is preserved under degree constraints by demonstrating a polynomial-time reduction from general Vertex Cover to Vertex Cover on graphs with maximum degree at most 3. The theoretical mechanism relies on "splitting" vertices of high degree into pairs connected by an edge, where the original vertex cover size bounds relate linearly (specifically $K$ vs. $K+L$) to the transformed instance, ensuring the reduction's runtime remains polynomial relative to the number of edges and new vertices introduced. Furthermore, it formalizes a second NP-completeness proof for Exact Cover by reducing from 3-Vertex Cover, utilizing auxiliary sets with unique identifiers ($X_1 \dots X_K$) to enforce exact cardinality constraints on subcollections while employing subset generation strategies restricted to bounded-degree graphs to maintain polynomial complexity.
This content establishes that NP-completeness is preserved under degree constraints by demonstrating a polynomial-time reduction from general Vertex Cover to Vertex Cover on graphs with maximum degre…