Critical Sets in Bipartite Graphs

TL;DR

This paper investigates critical independent sets in bipartite graphs, establishing key equalities and relationships among critical sets, core, ker, and diadem, thereby advancing understanding of their structural properties.

Contribution

It proves that in bipartite graphs, the critical difference equals the sum of maximum differences in each part, and that ker equals core, with a new relation involving ker, diadem, and maximum independent set size.

Findings

d_c(G) equals delta_0(A) plus delta_0(B)

ker(G) equals core(G)

|ker(G)| + |diadem(G)| = 2 * alpha(G)

Abstract

Let G=(V,E) be a graph. A set S is independent if no two vertices from S are adjacent, alpha(G) is the size of a maximum independent set, and core(G) is the intersection of all maximum independent sets. The number d(X)=|X|-|N(X)| is the difference of the set X, and d_{c}(G)=max{d(I):I is an independent set} is called the critical difference of G. A set X is critical if d(X)=d_{c}(G). For a graph G we define ker(G) as the intersection of all critical independent sets, while diadem(G) is the union of all critical independent sets. For a bipartite graph G=(A,B,E), with bipartition {A,B}, Ore defined delta(X)=d(X) for every subset X of A, while delta_0(A)=max{delta(X):X is a subset of A}. Similarly is defined delta_0(B). In this paper we prove that for every bipartite graph G=(A,B,E) the following assertions hold: d_{c}(G)=delta_0(A)+delta_0(B); ker(G)=core(G); |ker(G)|+|diadem(G)|=2*alpha(G).