Graphs with equal independence and matching number

Abstract

Extremal graphs are graphs which sit at the extremes. In simpler words for a class of graphs which satisfy a certain property, extremal graphs are the ones which exhibit a minimum or maximum of that property. Here, we take a look at a property which is exhibited by any graph in general; , where is the minimum degree of the graph, is the size of the maximum independent set, is the maximum degree, and is the size of the maximum matching of the graph. We first look at non-regular extremal graphs and regular extremal graphs (with degree 2 and 3) with respect to the above property as characterized by Mohr and Rautenbach. Later we try our hand at characterizing the regular extremal graphs using a general graph decomposition given jointly by Edmonds and Gallai. In doing so, we obtain a new proof for Mohr and Rautenbach’s characterization of 3-regular extremal graphs and we believe our approach can be easily adapted to characterize k-regular extremal graphs for values of k 3.