Decreasing the maximum degree of a graph

Abstract

For a non-empty graph G, let λ(G) be the smallest number of vertices that can be deleted from G so that the maximum degree of the resulting graph is smaller than the maximum degree Δ(G) of G. If G is regular, then λ(G) is the domination number γ(G) of G. We show that if 1≤k<r and c is a real number such that γ(H)≤c|V(H)| for every connected k-regular graph H with |V(H)|≥r, then λ(G)≤c|V(G)| for every connected graph G with Δ(G)=k and |V(G)|≥r. We in fact show thatλ(G)≤γ(H)|V(H)||V(G)| for an H explicitly constructed from G. Several bounds on λ(G) follow. Various problems motivated by the result are posed, and related results are obtained.We also provide a sharp bound on λ(G) that depends only on the vertices of largest degree. We call a vertex of largest degree a Δ-vertex. We call a Δ-vertex v solitary if no other Δ-vertex is of distance at most 2 from v. Let S(G) be the set of solitary Δ-vertices, and let T(G) be the set of non-solitary Δ-vertices. We show thatλ(G)≤|S(G)|+Δ(G)Δ(G)+1|T(G)|. The bound can be attained with T(G)≠∅.