Graphs in which all maximal bipartite subgraphs have the same order

TL;DR

This paper introduces well-bicovered graphs, where all maximal bipartite subgraphs share the same order, explores their properties, and characterizes specific classes including outerplanar graphs.

Contribution

It defines the concept of well-bicovered graphs, compares them to well-covered graphs, and characterizes certain subclasses and graph operations related to this property.

Findings

Examples of well-bicovered graphs provided

Characterizations of graphs with small or large bipartite number

Maximal outerplanar graphs that are well-bicovered characterized

Abstract

Motivated by the concept of well-covered graphs, we define a graph to be well-bicovered if every vertex-maximal bipartite subgraph has the same order (which we call the bipartite number). We first give examples of them, compare them with well-covered graphs, and characterize those with small or large bipartite number. We then consider graph operations including the union, join, and lexicographic and cartesian products. Thereafter we consider simplicial vertices and 3-colored graphs where every vertex is in triangle, and conclude by characterizing the maximal outerplanar graphs that are well-bicovered.