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.

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.

Equations

$$d(G)=\alpha (G)-\mu (G)=\left|\mathrm{core}(G)\right|-\left|N(\mathrm{core}(G))\right|,$$

$$N_1(C)=\{v:v\in V-V(C),N(v)\cap V(C)\ne \varnothing \},$$

$$n(G)=\alpha (G-e)+\mu (G-e)\le \alpha (G)+\mu (G)+1,$$

$$\alpha (G-e)-\alpha (G)+\mu (G-e)-\mu (G)=1,$$

$$-1\le \mu (G-e)-\mu (G)\le 0\le \alpha (G-e)-\alpha (G)\le 1.$$

$$n(G)-1=\alpha (G-e)+\mu (G-e)-1=\alpha (G)+\mu (G),$$

$$n(G)-1=\alpha (G)+\mu \left(G\right)\ge \alpha (G-y)+\mu \left(G-y\right)+1=n(G)-1+1=n(G),$$

$$\alpha \left(G\right)=\left|S_y-V(B_x)\right|+\left|S_y\cap V(B_x)\right|<\left|S_0\right|+\left|S_y\cap V(B_x)\right|=\left|S_0\cup \left(S_y\cap V(B_x)\right)\right|.$$

$$\left|S_y-V(B_x)\right|+\left|A\right|<\alpha (G)\le \alpha (G-B_x)+\alpha (B_x)=\left|S_y-V(B_x)\right|+\left|A\right|.$$

$$\alpha \left(G\right)=\left|S_{yz}-\left\{y\right\}\right|\le \left|\left(S_{yz}-\left\{y\right\}-\left(S_{yz}\cap V\left(B_x\right)\right)\right)\cup A\right|=\left|\left(S_{yz}-\left\{y\right\}-V\left(B_x\right)\right)\cup A\right|.$$

$$\mathrm{core}\left(B_x\right)$$

$$=(\bigcap \{S:S\in \Omega (G)\})\cap V\left(B_x\right)=\mathrm{core}\left(G\right)\cap V\left(B_x\right),$$

$$\left|\left(S_1-\left(S_1\cap V(B_x)\right)\right)\cup S_x\right|=\left|S_1-\left(S_1\cap V(B_x)\right)\right|+\left|S_x\right|$$

$$=\left|S_1\right|-\left|S_1\cap V(B_x)\right|+\alpha \left(B_x\right)\ge \alpha \left(G\right).$$

$$\left|W_a\right|=\left|S-V(C)\right|+\left|S_{ab}\right|$$

$$=\left|S\right|-\lfloor \left|V(C)\right|/2\rfloor +\lfloor \left|V(C)\right|/2\rfloor +1=1+\alpha \left(G\right),$$

$$\alpha (G)+1=\alpha (G-xy)>n\left(G-xy\right)/2=n(G)/2\ge$$

$$\mu (G-xy)=n(G)-\alpha (G-xy)=n(G)-\alpha (G)-1=\mu (G),$$

$$\alpha (G)-\mu (G)\le \left|\mathrm{core}(G)\right|-\left|N(\mathrm{core}(G))\right|.$$

$$\left|M_1\right|+\left|M_2\right|=\left|N(\mathrm{core}(G))\right|+\left|S-\mathrm{core}(G)\right|$$

$$=\left|N(\mathrm{core}(G))\right|+\alpha (G)-\left|\mathrm{core}(G)\right|\le \mu (G),$$

$$\alpha (G)-\mu (G)\le \left|\mathrm{core}(G)\right|-\left|N(\mathrm{core}(G))\right|=d(G)\le \alpha (G)-\mu (G)+1.$$

$$\left|A\right|-\left|N_G(A)\right|\le \left|A\right|-\left|N_H(A)\right|.$$

$$\alpha (G)-\mu (G)$$

$$=\alpha (H)-\mu (H)=\alpha (G)-\mu (G)+1.$$

$$\left|A\cap V\left(B_x\right)\right|-\left|N_{B_x}(A\cap V\left(B_x\right))\right|\le \left|\mathrm{core}(B_x)\right|-\left|N_{B_x}(\mathrm{core}(B_x))\right|=d\left(B_x\right)$$

$$\left|A\cap V\left(B_x\right)\right|-\left|N_G(A\cap V\left(B_x\right))\right|\le \left|A\cap V\left(B_x\right)\right|-\left|N_{B_x}(A\cap V\left(B_x\right))\right|$$

$$\left|\mathrm{core}(B_x)\right|-\left|N_{B_x}(\mathrm{core}(B_x))\right|=\left|\mathrm{core}(B_x)\right|-\left|N_G(\mathrm{core}(B_x))\right|\text{.}$$

$$\left|A\cap V\left(B_x\right)\right|-\left|N_G(A\cap V\left(B_x\right))\right|\le \left|\mathrm{core}(B_x)\right|-\left|N_G(\mathrm{core}(B_x))\right|.$$

$$A=\left(A\cap C\right)\cup \bigcup _{x\in N_1\left(C\right)}\left(A\cap V\left(B_x\right)\right)\text{ and }B=\left(A\cap C\right)\cup \bigcup _{x\in N_1\left(C\right)}\mathrm{core}(B_x).$$

$$d(G)=\sum _{x\in N_1\left(C\right)}d(B_x)=\sum _{x\in N_1\left(C\right)}\left(\left|\mathrm{core}(B_x)\right|-\left|N(\mathrm{core}(B_x))\right|\right)=\left|\mathrm{core}(G)\right|-\left|N(\mathrm{core}(G))\right|,$$

$$d(G)=\alpha (G)-\mu (G)=\left|\mathrm{core}(G)\right|-\left|N(\mathrm{core}(G))\right|.$$

$$d_G\left(S\right)=\left|S\right|-\left|N_G\left(S\right)\right|\le \left|S\right|-\left|N_H\left(S\right)\right|=d_H\left(S\right).$$

Full text

A graph $G$ is (i) unicyclic if it has a unique cycle, (ii) almost bipartite if it has only one odd cycle. It was conjectured in [15, 19] and validated in [1] that $d(G)=\alpha(G)-\mu(G)$ holds for every unicyclic non-König-Egerváry 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

[TABLE]

where by core$\left(G\right)$ we mean the intersection of all maximum independent sets.

Keywords: independent set, core, matching, critical set, critical difference, bipartite graph, König-Egerváry graph.

1 Introduction

Throughout this paper $G=(V,E)$ is a finite, undirected, loopless graph without multiple edges, with vertex set $V=V(G)$ of cardinality $n\left(G\right)$, and edge set $E=E(G)$ of size $m\left(G\right)$. If $X\subset V$, then $G[X]$ is the subgraph of $G$ spanned by $X$. By $G-W$ we mean the subgraph $G[V-W]$, if $W\subset V(G)$. For $F\subset E(G)$, by $G-F$ we denote the partial subgraph of $G$ obtained by deleting the edges of $F$, and we use $G-e$, if $W$ $=\{e\}$. If $A,B$ $\subset V$ and $A\cap B=\emptyset$, then $(A,B)$ stands for the set $\{e=ab:a\in A,b\in B,e\in E\}$. The neighborhood of a vertex $v\in V$ is the set $N(v)=\{w:w\in V$ and $vw\in E\}$, and $N(A)=\cup\{N(v):v\in A\}$, $N[A]=A\cup N(A)$ for $A\subset V$. By $C_{n},K_{n}$ we mean the chordless cycle on $n\geq$ $4$ vertices, and respectively the complete graph on $n\geq 1$ vertices.

Let us define the trace of a family $\mathcal{F}$of sets on the set $X$ as $\mathcal{F}|_{X}=\{F\cap X:F\in\mathcal{F}\}$.

A set $S$ of vertices is independent if no two vertices from $S$ are adjacent, and an independent set of maximum size will be referred to as a maximum independent set. The *independence number of $G$, denoted by $\alpha(G)$, is the cardinality of a maximum independent set *of $G$.

Let $\Omega(G)=\{S:S$ is a maximum independent set of $G\}$, core$(G)=\cap\{S:S\in\Omega(G)\}$ [12], and corona$(G)=\cup\{S:S\in\Omega(G)\}$ [4]. An edge $e\in E(G)$ is $\alpha$-critical whenever $\alpha(G-e)>\alpha(G)$. Notice that $\alpha(G)\leq\alpha(G-e)\leq\alpha(G)+1$ holds for each edge $e$.

The number $d(X)=\left|X\right|-\left|N(X)\right|$, $X\subseteq V(G)$, is called the difference of the set $X$. The number $d(G)=\max\{d(X):X\subseteq V\}$ is called the critical difference of $G$, and a set $U\subseteq V(G)$ is critical if $d(U)=d(G)$ [27]. The number $id(G)=\max\{d(I):I\in\mathrm{Ind}(G)\}$ is called the critical independence difference of $G$. If $A\subseteq V(G)$ is independent and $d(A)=id(G)$, then $A$ is called critical independent [27]. Clearly, $d(G)\geq id(G)$ is true for every graph $G$.

Theorem 1.1

[27]** The equality $d(G)$ $=id(G)$ holds for every graph $G$.

For a graph $G$, let denote $\mathrm{\ker}(G)=\bigcap\left\{S:S\text{ {is a critical independent set}}\right\}$. It is known that  $\mathrm{\ker}(G)\subseteq\mathrm{core}(G)$ is true for every graph [17], while the equality holds for bipartite graphs [21].

A matching (i.e., a set of non-incident edges of $G$) of maximum cardinality $\mu(G)$ is a maximum matching, and a perfect matching is one covering all vertices of $G$. An edge $e\in E(G)$ is $\mu$-*critical *provided $\mu(G-e)<\mu(G)$.

Theorem 1.2

For any graph $G$, the following assertions are true:

(i)* [14] no $\alpha$-critical edge has an endpoint in $N[\mathrm{core}(G)]$;*

(ii)* [4] there is a matching from $S-\mathrm{core}(G)$ into $\mathrm{corona}(G)-S$, for each $S\in\Omega(G)$;*

(iii) **[12]** if $G$ is a connected bipartite graph with $n\left(G\right)\geq 2$, then \alpha(G)>n\left(G\right)/2\if and only if$\ \left|\mathrm{core}(G)\right|\geq 2$.

It is well-known that $\lfloor n\left(G\right)/2\rfloor+1\leq\alpha(G)+\mu(G)\leq n\left(G\right)$ hold for every graph $G$. If $\alpha(G)+\mu(G)=n\left(G\right)$, then $G$ is called a König-Egerváry graph [5, 25]. Various properties of König-Egerváry graphs are presented in [2, 3, 9, 10, 13, 20]. It is known that every bipartite graph is a König-Egerváry* *graph [8, 7]. This class includes also non-bipartite graphs (see, for instance, the graph $G$ in Figure 1).

Theorem 1.3

If $G$ is of a König-Egerváry graph, then

(i)* [13] every maximum matching matches $N($core$(G))$ into core$(G)$;*

(ii) **[16]** $d(G)=\left|\mathrm{core}(G)\right|-\left|N(\mathrm{core}(G))\right|=\alpha(G)-\mu(G)$.

The graph $G$ is unicyclic if it has a unique cycle. We call a graph $G$ (edge) almost bipartite if it has a unique odd cycle, denoted by $C=\left(V(C),E\left(C\right)\right)$. Since $C$ is unique, there is no other cycle of $G$ sharing vertices with $C$. Let

[TABLE]

and $B_{x}=(V_{x},E_{x})$ be the bipartite connected* *subgraph of $G-xy$ containing $x$, where $x\in N_{1}(C),y\in V(C)$. Clearly, every unicyclic graph with an odd cycle is almost bipartite.

The smallest number of edges that have to be deleted from a graph to obtain a bipartite graph is called the bipartite edge frustration of $G$ and denoted by $\varphi\left(G\right)$ [6, 26]. Thus, $G$ is an almost bipartite graph whenever $\varphi\left(G\right)=1$.

In this paper we analyze the relationship between several parameters of a almost bipartite graph $G$, namely, $\mathrm{core}(G)$, $d(G)$, $\alpha\left(G\right)$, and $\mu\left(G\right)$.

2 Results

Lemma 2.1

If $G$ is a almost bipartite graph, then there is an edge $e\in E\left(C\right)$, such that $\mu(G-e)=\mu(G)$.

Proof. For every pair of edges, consecutive on $C$, only one of them may belong to every maximum matching of $G$. In other words, at most one of the edges could be $\mu$-critical.

Notice that $\alpha(G)\leq\alpha(G-e)\leq\alpha(G)+1$ holds for each edge $e$. Every edge of the unique odd cycle could be $\alpha$-critical; e.g., the graph $G$ from Figure 2.

Lemma 2.2

[18]** For every bipartite graph $H$, a vertex $v\in\mathrm{core}(H)$ if and only if there exists a maximum matching that does not saturate $v$.

Lemma 2.2 fails for non-bipartite König-Egerváry graphs; e.g., every maximum matching of the graph $G$ from Figure 1 saturates $c\in$ core$(G)=\{a,b,c\}$.

Lemma 2.3

If $G$ is a almost bipartite graph, then $n(G)-1\leq\alpha(G)+\mu(G)\leq n(G)$.

Proof. If $e=xy\in E(C)$, then $G-e$ is bipartite, and hence, $\alpha(G-e)+\mu(G-e)=n(G)$. Clearly, $\alpha(G-e)\leq\alpha(G)+1$, while $\mu(G-e)\leq\mu(G)$. Consequently, we get that

[TABLE]

which leads to $n(G)-1\leq\alpha(G)+\mu(G)$. The inequality $\alpha(G)+\mu(G)\leq n(G)$ is true for every graph $G$.

Lemma 2.4

Let $G$ be a almost bipartite graph. Then $n(G)-1=\alpha(G)+\mu(G)$ if and only if each edge of its unique odd cycle is $\alpha$-critical.

Proof. Assume that $n(G)-1=\alpha(G)+\mu(G)$. For each $e\in E(C)$, $G-e$ is bipartite, and then we have

[TABLE]

which implies $\mu(G-e)=\mu(G)$ and $\alpha(G-e)=\alpha(G)+1$, since

[TABLE]

In other words, every $e\in E(C)$ is $\alpha$-critical.

Conversely, let us choose $e\in E\left(C\right)$ satisfying $\mu(G-e)=\mu(G)$. By Lemma 2.1 such an edge exists. Since $e$ is $\alpha$-critical, and $G-e$ is bipartite, we infer that

[TABLE]

and this completes the proof.

Lemma 2.5

Let $G$ be a almost bipartite graph. If there is some $x\in N_{1}(C)$, such that $x\in\mathrm{core}(B_{x})$, then $G$ is a König-Egerváry graph.

Proof. Let $x\in\mathrm{core}(B_{x})$, $y\in N\left(x\right)\cap V(C)$, and $z\in N\left(y\right)\cap V(C)$. Suppose, to the contrary, that $G$ is not a König-Egerváry graph. By* *Lemma 2.3 and Lemma 2.4, the edge $yz$ is $\alpha$-critical. Since $y\notin\mathrm{core}(G)$, it follows that $\alpha(G)=\alpha(G-y)$. By Lemma 2.2 there exists a maximum matching $M_{x}$ of $B_{x}$ not saturating $x$. Combining $M_{x}$ with a maximum matching of $G-y-B_{x}$ we get a maximum matching $M_{y}$ of $G-y$. Hence $M_{y}\cup\left\{xy\right\}$ is a matching of $G$, which results in $\mu\left(G\right)\geq\mu\left(G-y\right)+1$. Consequently, using Lemma 2.4 and having in mind that $G-y$ is a bipartite graph of order $n(G)-1$, we get the following contradiction

[TABLE]

that completes the proof.

Theorem 2.6

If $G$ is a almost bipartite non-König-Egerváry graph, then $\Omega\left(G\right)|_{V\left(B_{x}\right)}=\Omega\left(B_{x}\right)$.

Proof. First, one has to prove that every maximum independent set of $B_{x}$ may be enlarged to some maximum independent set of $G$.

Let $A\in\Omega(B_{x})$, $y\in N\left(x\right)\cap V(C)$, and $z\in N\left(y\right)\cap V(C)$. According to Lemma 2.4, the edge $yz$ is $\alpha$-critical. Hence, there exist $S_{y}\in\Omega(G)$, $S_{yz}\in$ $\Omega(G-yz)$, such that $y\in S_{y}$ and $y,z\in S_{yz}$.

Case 1. Assume that $x\notin A$.

If $\left|S_{y}-V(B_{x})\right|<\alpha(G-B_{x})$ and $S_{0}\in\Omega\left(G-B_{x}\right)$, then $S_{0}\cup\left(S_{y}\cap V(B_{x})\right)$ is independent in $G$ that causes the contradiction

[TABLE]

Therefore, we have $\left|S_{y}-V(B_{x})\right|=\alpha(G-B_{x})$. Then $A\cup\left(S_{y}-V\left(B_{x}\right)\right)\in\Omega(G)$, otherwise we get the following contradiction

[TABLE]

Case 2. Assume now that $x\in A$.

Then $\left|A\right|\geq\left|S_{yz}\cap V\left(B_{x}\right)\right|$. Hence

[TABLE]

Since the set $\left(S_{yz}-\left\{y\right\}-V\left(B_{x}\right)\right)\cup A$ is independent and its size is $\alpha\left(G\right)$ at least, it is also maximum independent, i.e., $\left(S_{yz}-\left\{y\right\}-V\left(B_{x}\right)\right)\cup A\in\Omega(G)$.

Second, it is left to prove that $S\cap V\left(B_{x}\right)\in\Omega\left(B_{x}\right)$ for every $S\in\Omega\left(G\right)$. Let $S\in\Omega\left(G\right)$, and suppose, to the contrary, that $A=S\cap V\left(B_{x}\right)\notin\Omega\left(B_{x}\right)$. Since, by Lemma 2.5, we have $x\notin\mathrm{core}(B_{x})$, we can change $A$ for some $B\in\Omega\left(B_{x}\right)$ not containing $x$. The set $\left(S-A\right)\cup B$ is independent, and $\left|\left(S-A\right)\cup B\right|=\left|S-A\right|+\left|B\right|>\left|S\right|=\alpha(G)$. This contradiction completes the proof.

Corollary 2.7

If $G$ is a connected almost bipartite non-König-Egerváry graph, then

(i)* $\mathrm{core}\left(G\right)=\bigcup\left\{\mathrm{core}\left(B_{x}\right):x\in N(V(C))-V(C)\right\}$;*

(ii)* core$(G)\cap N\left[V\left(C\right)\right]=\emptyset$.*

Proof. *(i) *By Theorem 2.6, we infer that:

[TABLE]

which clearly implies $\mathrm{core}\left(G\right)=\bigcup\left\{\mathrm{core}\left(B_{x}\right):x\in N(V(C))-V(C)\right\}$.

*(ii) Let *$ab\in E\left(C\right)$. By Lemma 2.4, the edge $ab$ is $\alpha$-critical. Hence there exist $S_{a},S_{b}\in\Omega\left(G\right)$, such that $a\in S_{a}$ and $b\in S_{b}$. Since $a\notin S_{b}$, it follows that $a\notin$ core$(G)$, and because $a\in S_{a}$, we infer that $N\left(a\right)\cap$ core$(G)=\emptyset$. Consequently, we obtain that core$(G)\cap N\left[V\left(C\right)\right]=\emptyset$.

The assertion in Corollary 2.7(i) may fail for connected unicyclic König-Egerváry graphs; for instance, $\mathrm{core}\left(G_{2}\right)\neq\left\{x,z\right\}=\bigcup\left\{\mathrm{core}\left(B_{x}\right):x\in N(V(C))-V(C)\right\}$, while $\mathrm{core}\left(G_{1}\right)=\bigcup\left\{\mathrm{core}\left(B_{x}\right):x\in N(V(C))-V(C)\right\}$, where $G_{1}$ and $G_{2}$ are from Figure 3.

Proposition 2.8

Let $G$ be a almost bipartite. Then the following assertions are equivalent:

(i)* $x\notin\mathrm{core}(B_{x})$, for every $x\in N_{1}(C)$;*

(ii)* there exists some $S\in\Omega(G)$, such that $S\cap N_{1}(C)=\emptyset$;*

(iii)* $n(G)-1=\alpha(G)+\mu(G)$, i.e., $G$ is not a König-Egerváry graph.*

Proof. *(i) *$\Rightarrow$ *(ii) *Let $x\in N_{1}(C)$ and assume that there is $S_{1}\in\Omega(G)$, such that $x\in S_{1}$. Since $x\notin\mathrm{core}(B_{x})$, there exists some $S_{x}\in\Omega(B_{x})$, such that $x\notin S_{x}$. Hence we infer that $\left|S_{1}\cap V(B_{x})\right|\leq\alpha\left(B_{x}\right)=\left|S_{x}\right|$, $\left(S_{1}-\left(S_{1}\cap V(B_{x})\right)\right)\cup S_{x}$ is independent in $G$, and then

[TABLE]

Therefore $S_{2}=\left(S_{1}-\left(S_{1}\cap V(B_{x})\right)\right)\cup S_{x}\in\Omega(G)$, and $x\notin S_{2}$.

In this way, adding more vertices belonging to $N_{1}(C)$, one can build some $S\in\Omega(G)$, such that $S\cap N_{1}(C)=\emptyset$.

*(ii) *$\Rightarrow$ *(iii) *We have that $\left|S\cap V(C)\right|=\left\lfloor\left|V(C)\right|/2\right\rfloor$, because $S\cap N_{1}(C)=\emptyset$.

Let $ab\in E(C)$. Since $C$ is a chordless odd cycle, say $C=C_{2k+1},k\geq 1$, the edge $ab$ is $\alpha$-critical in $C$, i.e., there is $S_{ab}\in\Omega\left(C-ab\right)$, such that $a,b\in S_{ab}$ and $\left|S_{ab}\right|=k+1$.

Then, $W_{a}=(S-V(C))\cup S_{ab}$ is an independent set in $G-ab$, with

[TABLE]

which implies that the edge $ab$ is $\alpha$-critical in $G$. Since $ab$ was an arbitrary edge on $C$, it follows that every edge of $C$ is $\alpha$-critical in $G$. By Lemma* *2.4, it follows that $n(G)-1=\alpha(G)+\mu(G)$.

*(iii) *$\Rightarrow$ *(i) *It follows by Lemma 2.5.

Combining Lemma 2.5 and Proposition 2.8, we get the following.

Corollary 2.9

A almost bipartite graph is a König-Egerváry graph if and only if there is some $B_{x}$ such that $x\in\mathrm{core}(B_{x})$.

Theorem 2.10

Let $G$ be a connected almost bipartite graph. Then the following assertions are true:

(i)* $\mu(G)\leq\alpha(G)$;*

(ii)* there exists a matching from $N(\mathrm{core}(G))$ into $\mathrm{core}(G)$;*

(iii)* there is a maximum matching of $G$ that matches $N(\mathrm{core}(G))$ into $\mathrm{core}(G)$.*

Proof. If $G$ is a König-Egerváry graph, then (i) follows from the definition and the fact that $\mu\left(G\right)\leq n(G)/2$, while (ii), (ii) are true, by Theorem 1.3(i).

For the rest of the proof, we suppose that $G$ is not a König-Egerváry graph.

(i) By Lemma 2.3, we have $n(G)-1=\alpha(G)+\mu(G)$. According to Lemma 2.4, $\alpha(G-xy)=\alpha(G)+1$ holds for each edge $xy\in E(C)$. Consequently, we get that $x,y\in$ $\mathrm{core}(G-xy)$. Since $G-xy$ is bipartite, Theorem 1.2(iii) ensures that

[TABLE]

which results in $\alpha(G)\geq\mu(G)$.

*(ii) *If $\mathrm{core}(G)=\emptyset$, then the conclusion is clear.

Assume that $\mathrm{core}(G)\neq\emptyset$. By Theorem 1.3(i), in each $B_{x}$ there is a matching $M_{x}$ from $N(\mathrm{core}(B_{x}))$ into $\mathrm{core}(B_{x})$. By Theorem 1.2(i), it follows that $V(C)\cap N\left[\mathrm{core}(G)\right]=\emptyset$. Taking into account Corollary 2.7(i), we see that the union of all these matchings $M_{x}$ gives a matching from $N(\mathrm{core}(G))$ into $\mathrm{core}(G)$.

(iii) Let $M$ be a maximum matching of $G$ and $M_{1}$ be a matching from $N(\mathrm{core}(G))$ into $\mathrm{core}(G)$, that exists by Part (ii). The matching $M$ must saturate $N(\mathrm{core}(G))$, because otherwise it can be enlarged with edges from $M_{1}$. Hence, all the edges of $M$ saturating $N(\mathrm{core}(G))$ can be replaced by the edges of $M_{1}$, and the resulting matching is a maximum matching of $G$ that matches $N(\mathrm{core}(G))$ into $\mathrm{core}(G)$.

The almost bipartite graph $G$ from Figure 2 has $M_{1}=\{uv,cx,dt,wy\}$ and $M_{2}=\{uv,ac,dt,wy\}$ as maximum matchings, but only $M_{2}$ matches $N($core$(G))=\{c\}$ into core$(G)=\{a,b\}$. Notice that $G$ is not a König-Egerváry graph.

Proposition 2.11

If there is a matching from $N(\mathrm{core}(G))$ into $\mathrm{core}(G)$, then

[TABLE]

Proof. Let $M_{1}$ be a matching from $N(\mathrm{core}(G))$ into $\mathrm{core}(G)$. According to Theorem 1.2(ii), there is a matching, say $M_{2}$, from $S-\mathrm{core}(G)$ into $\mathrm{corona}(G)-S$. Consequently, we get that

[TABLE]

and this completes the proof.

Theorem 2.12

If $G$ is a connected almost bipartite graph, then

[TABLE]

Proof. If $G$ is a König-Egerváry graph, the result is true by Theorem 1.3(ii).

Otherwise, let $e\in E(C)$. Then $H=G-e$ is a bipartite graph, and by Lemma 2.4, we get that $\alpha(H)=\alpha(G)+1$ and $\mu(H)=\mu(G)$. For every $A\subseteq V(G)$, it follows that $\left|N_{H}(A)\right|\leq\left|N_{G}(A)\right|$, which implies

[TABLE]

Hence, using Theorem 2.10, Proposition 2.11, and Theorem 1.3(ii), we obtain

[TABLE]

Let $A$ be some critical independent set of $G$. By Theorem 1.3(ii), we have

[TABLE]

for every $x\in N_{1}\left(C\right)$. It is clear that

[TABLE]

Since, by Proposition 2.8(i), $x\notin\mathrm{core}(B_{x})$ for every $x\in N_{1}\left(C\right)$, we have

[TABLE]

Thus, by Theorem 1.3(ii), it follows

[TABLE]

Consequently, we infer that $\left|A\right|-\left|N(A)\right|\leq\left|B\right|-\left|N(B)\right|\leq d(G)$, where

[TABLE]

Using Corollary 2.7(i), we deduce that

[TABLE]

which completes the proof.

Theorem 2.13

[1]** If $G$ is unicyclic and non-König-Egerváry, then $d(G)=\alpha(G)-\mu(G)$.

Lemma 2.14

[11]** Every connected bipartite graph has a spanning tree with the same independence number.

Theorem 2.15

If $G$ is an almost bipartite non-König-Egerváry graph, then

[TABLE]

Proof. Case 1. $G$ is connected.

By Lemma 2.14, every bipartite subgraph $B_{x}$ of $G$ has a spanning tree $T_{x}$, having the same independence number, and hence, the same matching number, i.e., $\alpha\left(T_{x}\right)=\alpha\left(B_{x}\right)$ and $\mu\left(T_{x}\right)=\mu\left(B_{x}\right)$.

Consequently, $\Omega\left(B_{x}\right)\subseteq\Omega\left(T_{x}\right)$, which gives $\mathrm{core}\left(T_{x}\right)\subseteq\mathrm{core}\left(B_{x}\right)$. By Theorem 2.6, we have that $\Omega\left(G\right)|_{V\left(B_{x}\right)}=\Omega\left(B_{x}\right)$.

Let $H$ be the graph obtained from $G$ by substituting every $B_{x}$ with an appropriate $T_{x}$. Thus $H$ is a connected unicyclic graph, having $C$ as its unique cycle.

Since $G$ is a non-König-Egerváry graph, Proposition 2.8(i) implies $x\notin\mathrm{core}(B_{x})$, for every $x\in N_{1}(C)$. Therefore, $x\notin\mathrm{core}(T_{x})$, for every $x\in N_{1}(C)$.

Claim 1. $d(G)\leq d(H)$.

Every independent set $S$ of $G$ is independent in $H$ as well, while $N_{H}\left(S\right)\subseteq N_{G}\left(S\right)$. Hence,

[TABLE]

Thus, $d(G)\leq d(H)$.

Claim 2. $\alpha(G)=\alpha(H)$.

Since $G$ and $H$ have the same vertex sets and $E\left(H\right)\subseteq E\left(G\right)$, we get that $\alpha(G)\leq\alpha(H)$.

By Proposition 2.8(ii), there exists some $A\in\Omega(G)$, such that $A\cap N_{1}(C)=\emptyset$. Hence,

[TABLE]

Clearly, $A$ is an independent set in $H$ as well.

Let $S\in\Omega(H)$. Then, $\left|S\cap V\left(C\right)\right|\leq\left\lfloor V\left(C\right)/2\right\rfloor=\left|A\cap V\left(C\right)\right|$, and also

[TABLE]

Thus

[TABLE]

In conclusion, we get that $\alpha(G)=\alpha(H)$.

Claim 3. $\mu(G)=\mu(H)$.

Along the lines of the proof of Claim 2, we know that there exists a set $A\in\Omega(H)$, such that $A\cap N_{1}(C)=\emptyset$. Therefore, Proposition 2.8 implies that $H$ is a non-König-Egerváry graph. Hence,

[TABLE]

By Claim 2, it means that $\mu(G)=\mu(H)$.

Claim 4. $d(G)=\alpha(G)-\mu(G)$.

By Claim 2, Claim 3, Theorem 2.12, Claim 1, and Theorem 2.13, we finally obtain the following:

[TABLE]

which completes the proof.

Case 2. $G$ is disconnected.

Clearly, $G=G_{1}\cup G_{2}$, where $G_{1}$ is the connected component of $G$ containing the unique odd cycle, and $G_{2}$ is a nonempty bipartite graph. By Case 1,

[TABLE]

while Theorem 1.3*(ii) *implies

[TABLE]

Since

[TABLE]

we conclude with

[TABLE]

as required.

3 Conclusions

It is known that for every graph $\max\{0,\alpha(G)-\mu(G)\}\leq d(G)$ [17, 23, 24], while $\left|\mathrm{core}(G)\right|-\left|N(\mathrm{core}(G))\right|\leq d(G)$ by definition of $d(G)$.

By Theorems 1.3, 2.15 $d(G)=\alpha(G)-\mu(G)=\left|\mathrm{core}(G)\right|-\left|N(\mathrm{core}(G))\right|$ for both König-Egerváry graphs and almost bipartite graphs. Otherwise, every relation between $\alpha(G)-\mu(G)$ and $\left|\mathrm{core}(G)\right|-\left|N(\mathrm{core}(G))\right|$ is possible. For instance, the non-König-Egerváry graphs from Figure 4 satisfy

[TABLE]

The opposite direction of the displayed inequality may be found in $G_{3}=K_{2n}-e,n\geq 3$, where

[TABLE]

Another example reads as follows:

[TABLE]

where $G$ is from Figure 5.

Problem 3.1

Characterize graphs enjoying $d(G)=\alpha(G)-\mu(G)=\left|\mathrm{core}(G)\right|-\left|N(\mathrm{core}(G))\right|$.