# On the Critical Difference of Almost Bipartite Graphs

## TL;DR

This paper investigates the properties of almost bipartite graphs, establishing bounds on their independence and matching numbers, characterizing their structure, and relating the critical difference to the core of the graph.

## Contribution

It proves that for almost bipartite graphs, the sum of independence and matching numbers is either n-1 or n, and characterizes these cases, extending understanding of their critical difference.

## Key findings

- For almost bipartite graphs, lpha(G)+\u00b5(G) is either n(G)-1 or n(G).
- The critical difference d(G) equals | ext{core}(G)| - |N( ext{core}(G))| for these graphs.
- Characterizations of graphs achieving each sum value are provided.

## Abstract

A set $S\subseteq V$ is \textit{independent} in a graph $G=\left( V,E\right) $ if no two vertices from $S$ are adjacent. The \textit{independence number} $\alpha(G)$ is the cardinality of a maximum independent set, while $\mu(G)$ is the size of a maximum matching in $G$. If $\alpha(G)+\mu(G)$ equals the order of $G$, then $G$ is called a \textit{K\"{o}nig-Egerv\'{a}ry graph }\cite{dem,ster}. The number $d\left( G\right) =\max\{\left\vert A\right\vert -\left\vert N\left( A\right) \right\vert :A\subseteq V\}$ is called the \textit{critical difference} of $G$ \cite{Zhang} (where $N\left( A\right) =\left\{ v:v\in V,N\left( v\right) \cap A\neq\emptyset\right\} $). It is known that $\alpha(G)-\mu(G)\leq d\left( G\right) $ holds for every graph \cite{Levman2011a,Lorentzen1966,Schrijver2003}. In \cite{LevMan5} it was shown that $d(G)=\alpha(G)-\mu(G)$ is true for every K\"{o}nig-Egerv\'{a}ry graph. A graph $G$ is \textit{(i)} \textit{unicyclic} if it has a unique cycle, \textit{(ii)} \textit{almost bipartite} if it has only one odd cycle. It was conjectured in \cite{LevMan2012a,LevMan2013a} and validated in \cite{Bhattacharya2018} that $d(G)=\alpha(G)-\mu(G)$ holds for every unicyclic non-K\"{o}nig-Egerv\'{a}ry graph $G$. In this paper we prove that if $G$ is an almost bipartite graph of order $n\left( G\right) $, then $\alpha(G)+\mu(G)\in\left\{ n\left( G\right) -1,n\left( G\right) \right\} $. Moreover, for each of these two values, we characterize the corresponding graphs. Further, using these findings, we show that the critical difference of an almost bipartite graph $G$ satisfies \[ d(G)=\alpha(G)-\mu(G)=\left\vert \mathrm{core}(G)\right\vert -\left\vert N(\mathrm{core}(G))\right\vert , \] where by \textrm{core}$\left( G\right) $ we mean the intersection of all maximum independent sets.