Realizability of \(p\)-Point, \(q\)-Line Graphs with Prescribed Minimum Degree and Line Connectivity

Abstract

It is an established fact that some graph-theoretic extremal questions play an important part in the investigation of communication network vulnerability. Questions concerning the realizability of graph invariants are generalizations of these extremal problems. We define a \((p, q, \lambda, \delta)\) graph as a graph having \(p\) points, \(q\) lines, line connectivity \(\lambda\) and minimum degree \(\delta\). An arbitrary quadruple of integers \((a, b, c, d)\) is called \((p, q, \lambda, \delta)\) realizable if there is a \((p, q, \lambda, \delta)\) graph with \(p = a, q = b, \lambda = c\), and \(\delta = d\). Inequalities representing necessary and sufficient conditions for a quadruple to be \((p, q, \lambda, \delta)\) realizable are derived. In recent papers, the author gave necessary and sufficient conditions for \((p, q, \kappa, \Delta), (p, q, \lambda, \Delta), (p, q, \delta, \Delta)\) and \((p, q, \kappa, \delta)\) realizability, where \(\Delta\) denotes the maximum degree for all points in a graph and \(\lambda\) denotes the point connectivity of a graph. Boesch and Suffel gave the solutions for \((p, q, \kappa), (p, q, \lambda), (p, q, \delta), (p, \Delta, \delta, \lambda)\) and \((p, \Delta, \delta, \kappa)\) realizability in earlier manuscripts.