Realizability of p-point graphs with prescribed minimum degree, maximum degree, and point-connectivity

Abstract

In a recent paper, we gave a generalization of extremal problems involving certain graph-theoretic invariants. In that work, we defined a ( p, Δ, δ, λ) graph as a graph having p points, maximum degree Δ, minimum degree δ, and line-connectivity λ. An arbitrary quadruple of integers ( a, b, c, d) was called ( p, Δ, δ, λ) realizable if there is a ( p, Δ, δ, λ) graph with p = a, Δ = b, δ = c, and λ = d. In this work, we consider the more difficult case of ( p, Δ, δ, κ) realizability, where κ is the point-connectivity. Necessary and sufficient conditions for a quadruple to be ( p, Δ, δ, κ) realizable are derived.