Moving to Extremal Graph Parameters

TL;DR

This paper introduces a technique using compression and consolidation moves to determine extremal graphs for various parameters within graphs of fixed size, simplifying the process of finding minima and maxima.

Contribution

It develops a novel method involving graph transformations to efficiently identify extremal graphs for multiple parameters, advancing the understanding of graph optimization problems.

Findings

Minimizes the number of edges in line graphs and acyclic orientations.

Maximizes the number of cliques and forests.

Provides a unified approach for different graph parameters.

Abstract

Which graphs, in the class of all graphs with given numbers n and m of edges and vertices respectively, minimizes or maximizes the value of some graph parameter? In this paper we develop a technique which provides answers for several different parameters: the numbers of edges in the line graph, acyclic orientations, cliques, and forests. (We minimize the first two and maximize the third and fourth.) Our technique involves two moves on the class of graphs. A compression move converts any graph to a form we call fully compressed: the fully compressed graphs are split graphs in which the neighbourhoods of points in the independent set are nested. A second consolidation move takes each fully compressed graph to one particular graph which we call H(n,m). We show monotonicity of the parameters listed for these moves in many cases, which enables us to obtain our results fairly simply. The paper concludes with some open problems and future directions.