Upper-embeddability and the decycling number of connected 4-regular graphs: Upper-embeddability and the decycling number of connected 4-regular graphs - Long, Shude; Cai, Junliang (Graphs and combinatorics, 2025)
The decycling number of a graph G , denoted by ∇ ( G ) , is the smallest number of vertices whose removal results in an acyclic subgraph of G . A decycling set S of G with ∇ ( G ) vertices is said to be a ∇ -set. For any connected loopless 4-regular graph G on n vertices, it is shown that n + 1 3 ≤ ∇ ( G ) ≤ n + 1 2 , and presents a necessary and sufficient condition for the two equalities, respectively. Moreover, for any ∇ -set S of G , it is also shown that if G - S is a tree and ∇ ( G ) = n + 1 2 or n 2 , then G is upper-embeddable. Meanwhile, there exists a Xuong-tree T X of G such that vertices of S are leaves of T X .