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 \(\left| V\left( G\right) \right| =n\left( G\right) \), and edge set \(E=E(G)\) of size \(\left| E\left( G\right) \right| =m\left( G\right) \).

If \(X\subset V\), then G[X] is the subgraph of G induced by X. By \(G-v\) we mean the subgraph \(G[V-\left\{ v\right\} ]\), for \(v\in V\). The neighborhood of a vertex \(v\in V\) is the set \(N(v)=\{w:w\in V\) and \(vw\in E\}\). The neighborhood of \(A\subseteq V\) is \(N(A)=\{v\in V:N(v)\cap A\ne \emptyset \}\), and \(N\left[ A\right] =A\cup N(A)\), or \(N_{G}(A)\) and \(N_{G}\left[ A\right] \), if we specify the graph. If \(A,B\subset V\) are disjoint, then \(\left( A,B\right) =\left\{ ab:ab\in E,a\in A,b\in B\right\} \).

A set \(S\subseteq V(G)\) is independent if no two vertices from S are adjacent, and by \(\textrm{Ind}(G)\) we mean the family of all the independent sets of G. An independent set of maximum size is a maximum independent set of G, and \(\alpha (G)=\max \{\left| S\right| :S\in \textrm{Ind}(G)\}\). Let \(\Omega (G)\) be the family of all maximum independent sets, and \(\textrm{core}(G)=\bigcap \{S:S\in \Omega (G)\}\), while \(\xi (G)=\left| \textrm{core}(G)\right| \) [12]. A vertex \(v\in V(G)\) is \(\alpha \)-critical provided \(\alpha (G-v)<\alpha (G)\). Clearly, \(\textrm{core}(G)\) is the set of all \(\alpha \)-critical vertices of G. An edge \(e\in E(G)\) is \(\alpha \)-critical provided \(\alpha (G)<\alpha (G-e)\). Notice that there are graphs in which every edge is \(\alpha \)-critical (e.g., all \(C_{2k+1}\) for \(k\ge 1\)) or no edge is \(\alpha \)-critical (e.g., all \(C_{2k}\) for \(k\ge 2\)).

For \(X\subseteq V(G)\), the number \(\left| X\right| -\left| N(X)\right| \) is the difference of X, denoted d(X). The critical difference d(G) is \(\max \{d(I):I\in \textrm{Ind}(G)\}\). If \(A\in \textrm{Ind}(G)\) with \(d\left( A\right) =d(G)\), then A is a critical independent set [25]. For a graph G, let \(\textrm{MaxCritIndep}(G)=\{S:S\) is a maximum critical independent set\(\}\) [20], \(\mathrm {\ker }(G)\) be the intersection of all its critical independent sets and \(\varepsilon (G)=|\mathrm {\ker }(G)|\) [14, 16]. The critical independence number of a graph G, denoted as \(\alpha ^{\prime }\left( G\right) \), is the cardinality of a maximum critical independent set [10].

Theorem 1.1

(i) [4] Each critical independent set is included in some \(S\in \Omega (G)\).

(ii) [9] Every critical independent set is contained in some \(S\in \textrm{MaxCritIndep}(G)\).

(iii) [9] There is a matching from N(S) into S for every critical independent set S.

Theorem 1.2

[10] For any graph G, there is a unique set \(X\subseteq V(G)\) such that

(i) \(\alpha (G)=\alpha \left( G\left[ X\right] \right) +\alpha \left( G\left[ V\left( G\right) -X\right] \right) \);

(ii) \(X=N\left[ A\right] \) for every \(A\in \textrm{MaxCritIndep}(G)\);

(iii) \(G\left[ X\right] \) is a König-Egerváry graph.

(iv) \(G\left[ V\left( G\right) -X\right] \) has only \(\emptyset \) as a critical independent set.

Let \(diadem (G)=\bigcup \{S:S\) is a critical independent set in \(G\}\) [8], while \(\beta (G)=\left| diadem (G)\right| \).

Theorem 1.3

[14] If A and B are critical sets in a graph G, then \(A\cup B\) and \(A\cap B\) are critical as well.

A matching in a graph \(G=(V,E)\) is a set of edges \(M\subseteq E\) such that no two edges of M share a common vertex. A matching of maximum cardinality \(\mu (G)\) is a maximum matching, and a perfect matching is one saturating all vertices of G. Given a matching M in G, a vertex \(v\in V\) is called M-saturated if there exists an edge \(e\in M\) incident with v.

An edge \(e\in E(G)\) is \(\mu \)-critical provided \(\mu (G-e)<\mu (G)\). A vertex \(v\in V(G)\) is \(\mu \)-critical (essential) provided \(\mu (G-v)<\mu (G)\), i.e., v is M-saturated by every maximum matching M of G.

It is known that

$$ \alpha (G)+\mu (G)\le n\left( G\right) \le \alpha (G)+2\mu (G) $$

hold for every graph G [3]. If \(\alpha (G)+\mu (G)=n\left( G\right) \), then G is called a König-Egerváry graph [5, 7, 24]. If S is an independent set of a graph G and \(A=V\left( G\right) -S\), then we write \(G=S*A\). For instance, if \(E(G[A])=\emptyset \), then \(G=S*A\) is bipartite; if G[A] is a complete graph, then \(G=S*A\) is a split graph.

Theorem 1.4

For a graph G, the following properties are equivalent:

(i) G is a König-Egerváry graph;

(ii) [12] \(G=S*A\), where \(S\in \) \(\textrm{Ind}(G)\), \(\left| S\right| \ge \left| A\right| \), and \(\left( S,A\right) \) contains a matching M with \(\left| M\right| =\left| A\right| \);

(iii) [17] each maximum matching of G matches \(V\left( G\right) -S\) into S, for every \(S\in \Omega (G)\);

(iv) [14] every \(S\in \Omega (G)\) is critical.

The König deficiency of graph G is \(\kappa \left( G\right) =n\left( G\right) -\left( \alpha (G)+\mu (G)\right) \) [1]. Thus, a graph G is k-König-Egerváry if and only if \(\kappa \left( G\right) =k\) [21]. In particular, if \(\kappa \left( G\right) =0\), then G is a König-Egerváry graph. If \(\alpha (G)+\mu (G)=n\left( G\right) -1\), then G is a 1 -König-Egerváry graph. Thus, a graph G is 1 -König-Egerváry if and only if \(\kappa \left( G\right) =1\). A subfamily of 1-König-Egerváry graphs was characterized in [6], settling an open problem posed in [18].

For instance, both \(G_{1}\) and \(G_{2}\) from Figure 1 are 1-König-Egerváry graphs.

Fig. 1
figure 1

\(G_{1}-v_{1}\), \(G_{2}-e_{2}\) are König-Egerváry graphs, while \(G_{1}-v_{2}\) and \(G_{2}-e_{1}\) are not König-Egerváry graphs

Full size image

Definition 1.5

A graph G is called:

(i) a vertex almost König-Egerváry graph if G is not König-Egerváry, but there is a vertex \(v\in V(G)\), such that \(G-v\) is a König-Egerváry graph [11];

(ii) an edge almost König-Egerváry graph if G is not a König-Egerváry graph, but there exists an edge \(e\in E(G)\), such that \(G-e\) is a König-Egerváry graph.

For example, the graph \(G_{1}\) from Figure 1 is vertex almost König-Egerváry but not edge almost König-Egerváry, while \(G_{2}\) is edge almost König-Egerváry but not vertex almost König-Egerváry.

Lemma 1.6

[11] A graph G is a vertex almost König-Egerváry graph if and only if there is a vertex \(v\in V(G)\) such that \(G-v\) is a König-Egerváry graph, \(\alpha (G-v)=\alpha (G)\), and \( \mu (G-v)= \mu (G)\).

Clearly, every odd cycle \(C_{2k+1}\) is a vertex almost König-Egerváry graph, an edge almost König-Egerváry graph and an almost König-Egerváry graph.

Lemma 1.7

For a graph G, the following assertions hold:

(i) \(\alpha (G)\le \alpha (G-e)\le \alpha (G)+1\) for each \(e\in E(G)\);

(ii) \(\alpha (G)-1\le \alpha (G-v)\le \alpha (G)\) for every \(v\in V(G)\);

(iii) \(\mu (G)-1\le \mu (G-a)\le \mu (G)\) for each \(a\in V(G)\cup E(G)\).

Notice that: each edge of \(C_{2q+1}\) is \(\alpha \)-critical and non-\(\mu \)-critical; each vertex of \(C_{2q+1}\) is neither \(\alpha \)-critical and nor \(\mu \)-critical; each edge of \(C_{2q}\) is neither \(\alpha \)-critical and nor \(\mu \)-critical; each vertex of \(C_{2q}\) is \(\mu \)-critical, but non-\(\alpha \)-critical.

Using Lemma 1.7, one can easily get the following.

Corollary 1.8

The equality \(\alpha (G-e)+\mu (G-e)=\alpha (G)+\mu (G)\) holds if and only if either the edge \(e\in E(G)\) is both \(\alpha \)-critical and \(\mu \)-critical or the edge \(e\in E(G)\) is both non-\(\alpha \)-critical and non-\(\mu \)-critical.

For instance, consider the graphs from Figure 2:

  • \(\alpha (G_{1}-a)+\mu (G_{1}-a)=\alpha (G_{1})+\mu (G_{1})=6\), since a is both \(\alpha \)-critical and \(\mu \)-critical;

  • \(\alpha (G_{2}-u_{1})+\mu (G_{2}-u_{1})=\alpha (G_{2})+\mu (G_{2})+1=7\), since \(u_{1}\) is \(\alpha \)-critical and non-\(\mu \)-critical;

  • \(\alpha (G_{2}-u_{2})+\mu (G_{2}-u_{2})=\alpha (G_{2})+\mu (G_{2})=6\), since \(u_{2}\) is both non-\(\alpha \)-critical and non-\(\mu \)-critical;

  • \(\alpha (G_{3}-b)+\mu (G_{3}-b)=\alpha (G_{3})+\mu (G_{3})-1=4\), since b is non-\(\alpha \)-critical and \(\mu \)-critical.

Fig. 2
figure 2

Only \(G_{1}\) is a König-Egerváry graph

Full size image

Proposition 1.9

[13] In a König-Egerváry graph \(\alpha \)-critical edges are also \(\mu \)-critical, and they coincide in bipartite graphs.

Let \(\varrho _{v}\left( G\right) \) denote the number of vertices \(v\in V\left( G\right) \), such that \(G-v\) is a König-Egerváry graph, and \(\varrho _{e}\left( G\right) \) denote the number of edges \(e\in E\left( G\right) \) satisfying \(G-e\) is a König-Egerváry graph [22].

Theorem 1.10

[22] If G is a König-Egerváry graph, then

$$ \varrho _{v}\left( G\right) =n\left( G\right) -\xi \left( G\right) +\varepsilon \left( G\right) \text { and }\varrho _{e}\left( G\right) \le m\left( G\right) -\xi \left( G\right) +\varepsilon \left( G\right) . $$

In this paper, we characterize 1-König-Egerváry graphs, vertex (edge) almost König-Egerváry graphs, and present interrelationships between them. We also show that if G is a 1-König-Egerváry graph, then

$$ \varrho _{v}\left( G\right) \le n\left( G\right) +d\left( G\right) -\xi \left( G\right) -\beta (G). $$

As an application,we characterize the 1-König-Egerváry graphs that become König-Egerváry after deleting any vertex. In other words, for a 1-König-Egerváry graph G, the equality \(\varrho _{v}\left( G\right) =n\left( G\right) \) holds if and only if \(\xi \left( G\right) =0\), \(\beta \left( G\right) =0\) and \(\mu (G) < n(G)/2\).

2 Structural results

Definition 2.1

A set \(A\subseteq V(G)\) is supportive if either

(i) there is a vertex \(v\in V\left( G\right) -A\) and a matching from \(V\left( G\right) -A-v\) into A, or

(ii) there is an edge xy \(\in E\left( G-A\right) \) and a matching from \(V\left( G\right) -A-x-y\) into A.

For instance, consider the graphs in Figure 3: the set \(A=\left\{ a_{1},a_{2}\right\} \) is a supportive (maximum independent) set in \(G_{1}\), and \(B=\left\{ b_{1},b_{2}\right\} \) is a supportive set in \(G_{2} \).

Fig. 3
figure 3

Supportive sets: \(\left\{ a_{1},a_{2}\right\} \) and \(\left\{ b_{1},b_{2}\right\} \)

Full size image

The following finding gives a structural characterization of 1 -König- Egerváry graphs, similarly to some for König-Egerváry graphs [15, 17].

Theorem 2.2

Let G be a non-König-Egerváry graph. Then the following assertions are equivalent:

(i) G is a 1-König-Egerváry graph;

(ii) there exists a supportive maximum independent set in G;

(iii) every maximum independent set of G is supportive.

Proof

(i) \(\Rightarrow \) (iii) Assume that G is 1-König-Egerváry, S is an arbitrary maximum independent set, and M is a maximum matching. Hence,

$$ \mu \left( G\right) =n\left( G\right) -\alpha \left( G\right) -1=n\left( G\right) -\left| S\right| -1. $$

Let M contain \(b_{1}\) edges connecting S and \(V\left( G\right) -S\), while \(b_{2}\) edges connecting vertices from \(V\left( G\right) -S\). Thus \(\mu \left( G\right) =b_{1}+b_{2}\). Hence,

$$ n\left( G\right) -\alpha \left( G\right) =n\left( G\right) -\left| S\right| \ge b_{1}+2b_{2}=\mu \left( G\right) +b_{2}. $$

Therefore, we get

$$ 1=n\left( G\right) -\alpha \left( G\right) -\mu \left( G\right) \ge b_{2}. $$

Case 1. \(b_{2}=0\). Since \(\left| M\right| =\) \(\mu \left( G\right) =\)\(n\left( G\right) -\alpha \left( G\right) -1=\) \(\left| V\left( G\right) -S\right| -1\), we infer that M saturates all the vertices from \(V\left( G\right) -S\), except one, say \(v\in V\left( G\right) -S\), and then \(G-v\) is a König-Egerváry graph.

Case 2. \(b_{2}=1\). Since \(\left| M\right| -1=\) \(\mu \left( G\right) -1=\)\(n\left( G\right) -\alpha \left( G\right) -2=\)\(\left| V\left( G\right) -S\right| -2\), we conclude that M saturates all the vertices from \(V\left( G\right) -S\), and M contains exactly one edge that joins two vertices from \(V\left( G\right) -S\).

Thus S is supportive.

(iii) \(\Rightarrow \) (ii) Clear.

(ii) \(\Rightarrow \) (i) Let S be a supportive maximum independent set. Now, by the definition of a supportive set and the fact that \(S\in \Omega \left( G\right) \), we get \(\alpha (G)+\mu (G)=n\left( G\right) -1\), as required. \(\square \)

Corollary 2.3

A non-König-Egerváry graph G is 1 -König-Egerváry if and only if either there is a vertex \(v\in V\left( G\right) \) such that \(G-v\) is a König-Egerváry graph or there is an edge \(xy\in E\left( G\right) \) such that \(G-x-y\) is a König-Egerváry graph.

Proof

Suppose that G is 1-König-Egerváry. By Theorem 2.2, either there exists a vertex v such that \(G-v\) is König-Egerváry or there is an edge xy such that \(G-x-y\) is König-Egerváry.

Conversely, assume that \(G-v\) is a König-Egerváry graph, for some \(v\in V\left( G\right) \). Since G is not a König-Egerváry graph, we get that

$$ n\left( G\right) -1=\alpha (G-v)+\mu (G-v)\le \alpha (G)+\mu (G)\le n\left( G\right) -1\text {,} $$

which means that \(\alpha (G)+\mu (G)=n\left( G\right) -1\), i.e., G is 1-König-Egerváry.

Now, let \(xy\in E(G)\) be such that \(G-x-y\) is a König-Egerváry graph. Hence,

$$ n\left( G\right) -2=\alpha (G-x-y)+\mu (G-x-y)\le \alpha (G)+\mu (G)\le n\left( G\right) -1\text {.} $$

It is clear that \(\alpha (G)-\alpha (G-x-y)\ge 0\). On the other hand, \(\mu (G)-\mu (G-x-y)>0\), because \(xy\in E(G)\). Thus \(\alpha (G)+\mu (G)=n\left( G\right) -1\), as required. \(\square \)

For instance, the graphs from Figure 4 are 1 -König-Egerváry, as both \(G_{1}-x-y\) and \(G_{2}-v\) are König-Egerváry graphs.

Fig. 4
figure 4

Both \(G_{1}\) and \(G_{2}\) are 1-König-Egerváry graphs

Full size image

Clearly, Theorem 2.2 may be reformulated as follows.

Theorem 2.4

Let G be a non-König-Egerváry graph. Then G is a 1-König-Egerváry graph if and only if for every \(S\in \Omega \left( G\right) \), there is a vertex \(v\in V\left( G\right) -S\) and a matching from \(V\left( G\right) -S-\left\{ v\right\} \) into S, or there is an edge xy \(\in E\left( G-S\right) \) and a matching from \(V\left( G\right) -S-\left\{ x,y\right\} \) into S.

It is known that \(\mu (G)\le \alpha (G)\) is true for every König-Egerváry graph (Theorem 1.4(iii)). Consider the graphs in Figure 5 in order to see that for 1-König-Egerváry graphs the situation is different.

Fig. 5
figure 5

\(\mu (G_{1})=\alpha (G_{1})+1\), \(\mu (G_{2})=\alpha (G_{2} )\), and \(\mu (G_{3})\,<\alpha (G_{3})\)

Full size image

Theorem 2.5

If G is a 1-König-Egerváry graph, then

(i) \(\mu (G)\le \alpha (G)+1\);

(ii) \(\mu (G)=\alpha (G)+1\) if and only if G has a perfect matching;

(iii) \(\mu (G)<\alpha (G)\), whenever G has no perfect matchings and \(n\left( G\right) \) is even.

Proof

(i) According to Theorem 2.4, we distinguish between the following cases.

Case 1. There are a vertex \(v\in V\left( G\right) -S\) and a matching from \(V\left( G\right) -S-\left\{ v\right\} \) into S, where \(S\in \Omega \left( G\right) \).

By Theorem 1.4, it follows that \(\mu \left( G-v\right) =\mu \left( G\right) \) and \(\alpha \left( G-v\right) =\alpha \left( G\right) \). Since \(G-v\) is a König-Egerváry graph, we know that \(\mu \left( G-v\right) \le \alpha \left( G-v\right) \). Thus, \(\mu (G)\le \alpha (G)\le \alpha (G)+1\).

Case 2. There is an edge xy \(\in E\left( G-S\right) \) and a matching from \(V\left( G\right) -S-\left\{ x,y\right\} \) into S, where \(S\in \Omega \left( G\right) \).

By Theorem 1.4, it follows that \(\mu \left( G-\left\{ x,y\right\} \right) +1=\mu \left( G\right) \) and \(\alpha \left( G-\left\{ x,y\right\} \right) =\alpha \left( G\right) \). Since \(G-\left\{ x,y\right\} \) is a König-Egerváry graph, we obtain

$$ \mu \left( G-\left\{ x,y\right\} \right) \le \alpha \left( G-\left\{ x,y\right\} \right) \Leftrightarrow \mu \left( G\right) -1\le \alpha \left( G\right) , $$

as required.

(ii) If \(\mu (G)=\alpha (G)+1\), then \(n\left( G\right) =\mu (G)+\alpha (G)+1=2\mu (G)\), which means that G has a perfect matching.

Conversely, if G has a perfect matching, then \(2\mu (G)=n\left( G\right) =\mu (G)+\alpha (G)+1\), and this gives \(\mu (G)=\alpha (G)+1\).

(iii) If G has no perfect matchings, then \(\mu (G)<\frac{n\left( G\right) }{2}\). Hence,

$$ \frac{n\left( G\right) }{2}+\alpha (G)>\mu (G)+\alpha (G)=n\left( G\right) -1, $$

which means that \(\alpha (G)>\frac{n\left( G\right) }{2}-1\). Since \(\frac{n\left( G\right) }{2}\) is integer, we obtain \(\alpha (G)\ge \frac{n\left( G\right) }{2}\). Finally, we get \(\alpha (G)\ge \frac{n\left( G\right) }{2}>\mu (G)\), which completes the proof. \(\square \)

Theorem 2.6

If G is either an edge almost König-Egerváry graph or a vertex almost König-Egerváry graph, then G is a 1 -König-Egerváry graph as well.

Proof

Let \(e\in E(G)\) be such that \(G-e\) is a König-Egerváry graph. Since G is not a König-Egerváry graph, and clearly, \(\alpha (G)\le \alpha (G-e)\le \alpha (G)+1\) and \(\mu (G)\ge \mu (G-e)\), we obtain

$$ n\left( G\right) -1=n\left( G-e\right) -1=\alpha (G-e)-1+\mu (G-e)\le \alpha (G)+\mu (G)<n\left( G\right) \text {.} $$

Therefore, \(\alpha (G)+\mu (G)=n\left( G\right) -1\), i.e., G is a 1-König-Egerváry graph.

Let \(v\in V\left( G\right) \) be such that \(G-v\) is a König-Egerváry graph. Since G is not a König-Egerváry graph, we deduce that

$$ n\left( G\right) -1=n\left( G-v\right) =\alpha (G-v)+\mu (G-v)\le \alpha (G)+\mu (G)<n\left( G\right) \text {,} $$

which implies that \(\alpha (G)+\mu (G)=n\left( G\right) -1\), i.e., G is a 1-König-Egerváry graph. \(\square \)

Notice that there exist 1-König-Egerváry graphs that are neither edge almost König-Egerváry graphs, nor vertex almost König-Egerváry graphs; e.g., \(pK_{1}+K_{p+1}\), \(pK_{1} +K_{p+2}\).

In continuation of Lemma 1.6, we proceed with the following.

Theorem 2.7

(i) A graph G is vertex almost König-Egerváry if and only if it is 1-König-Egerváry and some \(v\in V(G)\) is neither \(\alpha \)-critical nor \(\mu \)-critical.

(ii) A graph G is edge almost König-Egerváry if and only if it is 1-König-Egerváry and some \(e\in E(G)\) is \(\alpha \)-critical and non-\(\mu \)-critical.

Proof

Since G is not a König-Egerváry graph, we know that \(\alpha (G)+\mu (G)<n\left( G\right) \).

(i) Assume that G is vertex almost König-Egerváry. By Theorem 2.6, G is also 1-König-Egerváry.

There exists \(v\in V\left( G\right) \) such that \(G-v\) is a König-Egerváry graph. According to Lemma 1.7(ii) and (iii), we get

$$ n\left( G\right) -1=\alpha (G-v)+\mu (G-v)\le \alpha (G)+\mu (G)<n\left( G\right) , $$

which implies that \(\alpha (G-v)=\alpha (G)\) and \(\mu (G-v)=\mu (G)\), and these mean that \(v\in V(G)\) is neither \(\alpha \)-critical nor \(\mu \)-critical.

The converse is clear, because \(\alpha (G-v)+\mu (G-v)=\alpha (G)+\mu (G)=n\left( G\right) -1\).

(ii) Suppose that G is vertex almost König-Egerváry. Then, by Theorem 2.6, G is also 1-König-Egerváry.

There exists \(xy\in E\left( G\right) \) such that \(G-xy\) is a König-Egerváry graph. According to Lemma 1.7(i) and (iii), we get

$$ n\left( G\right) =\alpha (G-xy)+\mu (G-xy)\le \alpha (G)+1+\mu (G)<n\left( G\right) +1, $$

which implies that \(\alpha (G-xy)=\alpha \left( G\right) +1\) and \(\mu (G-xy)=\mu \left( G\right) \), i.e., the edge xy is \(\alpha \)-critical and non-\(\mu \)-critical.

Conversely, we have that \(\alpha (G-xy)=\alpha \left( G\right) +1\), \(\mu (G-xy)=\mu \left( G\right) \), and \(\alpha (G)+\mu (G)=n\left( G\right) -1\), which ensures that

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

and this means that G is an edge almost König-Egerváry graph. \(\square \)

Recall that a graph is almost bipartite if it has a unique odd cycle [19, 23].

Lemma 2.8

[19] If G is an almost bipartite graph, then \(n(G)-1\le \alpha (G)+\mu (G)\le n(G)\).

Consequently, one may say that each almost bipartite graph is either a König-Egerváry graph or a 1-König-Egerváry graph.

Corollary 2.9

If G is an almost bipartite graph, then the following assertions are equivalent:

(i) G is a 1-König-Egerváry graph;

(ii) G is a vertex almost König-Egerváry graph;

(iii) G is an edge almost König-Egerváry graph.

Fig. 6
figure 6

A non almost bipartite 1-König-Egerváry graph

Full size image

By Theorem 2.7, G from Figure 6 is both a vertex and an edge almost König-Egerváry graph (since the vertex v is neither \(\alpha \)-critical nor \(\mu \)-critical, while the edge uv is \(\alpha \)-critical and non-\(\mu \)-critical).

3 The \(\varrho _{v}\)-inequality for 1-König-Egerváry graphs

Notice that for every \(n\ge 3\), the complete graph \(K_{n}\) is not König-Egerváry; the same is true for \(K_{n}-v\), whenever \(n\ge 4\). However, \(K_{3}-v\) is a König-Egerváry graph, while \(K_{4}-v\) is a 1-König-Egerváry graph, for every vertex v.

Theorem 3.1

If there is a vertex \(v\in V\left( G\right) \), such that \(G-v\) is a König-Egerváry graph, then G is either König-Egerváry or 1-König-Egerváry.

Proof

By definition, we know that

$$ \alpha (G-v)+\mu (G-v)=n\left( G-v\right) =n\left( G\right) -1. $$

Case 1. \(\alpha (G-v)=\alpha (G)\).

Thus \(\alpha (G)+\mu (G-v)=n\left( G\right) -1\). If \(\mu (G-v)=\mu (G)\), then \(\alpha (G)+\mu (G)=n\left( G\right) -1\), which means that G is a 1-König-Egerváry graph; otherwise, \(\mu (G-v)=\mu (G)-1\) and, consequently, \(\alpha (G)+\mu (G)=n\left( G\right) \), i.e., G is a König-Egerváry graph.

Case 2. \(\alpha (G-v)=\alpha (G)-1\).

Thus \(\alpha (G)-1+\mu (G-v)=n\left( G\right) -1\). If \(\mu (G-v)=\mu (G)\), then \(\alpha (G)+\mu (G)=n\left( G\right) \), which means that G is a König-Egerváry graph; otherwise, \(\mu (G-v)=\mu (G)-1\) and, consequently, \(\alpha (G)+\mu (G)=n\left( G\right) +1\), which is impossible, as \(\alpha (G)+\mu (G)\le n\left( G\right) \) for every graph G. \(\square \)

Corollary 3.2

If G is a König-Egerváry graph, then \(G+v\), where \(v\notin V\left( G\right) \) and \(N_{G+v}\left( v\right) =A\subseteq V\left( G\right) \), is either König-Egerváry or 1-König-Egerváry.

Consider the 1-König-Egerváry graphs from Figure 7. \(G_{1}-a\) is a König-Egerváry graph and the vertex a is neither \(\alpha \)-critical nor \(\mu \)-critical, while \(G_{1}-b\) is a not a König-Egerváry graph and the vertex b is \(\alpha \)-critical. The vertex x is both \(\alpha \)-critical and \(\mu \)-critical, and \(G_{2}-x\) is a not a König-Egerváry graph.

Fig. 7
figure 7

\(\alpha \)-critical vertices and \(\mu \)-critical vertices in 1-König-Egerváry graphs

Full size image

Theorem 3.3

Let G be a 1-König-Egerváry graph. Then \(G-v\) is König-Egerváry if and only if the vertex v is neither \(\alpha \)-critical nor \(\mu \)-critical.

Proof

According to the definition, we know that \(\alpha (G)+\mu (G)=n\left( G\right) -1\).

Assume that \(G-v\) is a König-Egerváry graph. Then we obtain

$$ \alpha \left( G-v\right) +\mu \left( G-v\right) =n\left( G-v\right) =n\left( G\right) -1=\alpha (G)+\mu (G). $$

Since both \(\alpha \left( G-v\right) \le \alpha \left( G\right) \) and \(\mu \left( G-v\right) \le \mu \left( G\right) \), we get \(\alpha \left( G-v\right) =\alpha \left( G\right) \) and \(\mu \left( G-v\right) =\mu \left( G\right) \), i.e., the vertex v is neither \(\alpha \)-critical nor \(\mu \)-critical.

Conversely, if v is neither \(\alpha \)-critical nor \(\mu \)-critical, then

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

which means that \(G-v\) is a König-Egerváry graph. \(\square \)

Theorem 3.4

Suppose that \(A\in \textrm{Ind}(G)\). If there is a matching from \(N_{G}(A)\) into A, then every matching from \(N_{G}(A)\) into A can be enlarged to a maximum matching of G, and every vertex of \(N_{G}(A)\) is \(\mu \)-critical.

Proof

Let M be a maximum matching of G, \(M\left( N_{G}(A)\right) \) be the vertices of \(N_{G}(A)\) that are saturated by M, and \(M_{1}\) be a matching from \(N_{G}(A)\) into A. If \(\left| M\left( N_{G}(A)\right) \right| <\left| N_{G}(A)\right| \), then \(M\cup M_{1}-M_{2}\), where \(M_{2}=\{ xy:xy\in M\text { and } x\in M\left( N_{G}(A)\right) \} \), is a matching of G larger than M, contradicting the maximality of M. Therefore, \(\left| M\left( N_{G}(A)\right) \right| =\left| N_{G}(A)\right| \), and \(M\cup M_{1}-M_{2}\) is a maximum matching of G that enlarges \(M_{2}\).

In other words, we have

$$ \mu \left( G\right) =\mu \left( G\left[ N_{G}\left[ A\right] \right] \right) +\mu \left( G\left[ V\left( G\right) -N_{G}\left[ A\right] \right] \right) , $$

which means that every matching from \(N_{G}(A)\) into A can be enlarged to a maximum matching of G.

Second, let \(v\in N_{G}(A)\) and \(H=G-v\). Then there is a matching from \(N_{H}(A)=\) \(N_{G}(A)-\left\{ v\right\} \) into A, and consequently,

$$\begin{aligned} & \mu \left( G-v\right) =\mu \left( H\right) =\mu \left( H\left[ N_{H}\left[ A\right] \right] \right) +\mu \left( H\left[ V\left( H\right) -N_{H}\left[ A\right] \right] \right) \\ & = \mu \left( G\left[ N_{G}\left[ A\right] -\left\{ v\right\} \right] \right) +\mu \left( G\left[ V\left( G\right) -N_{G}\left[ A\right] \right] \right) \\ & = \mu \left( G\left[ N_{G}\left[ A\right] \right] \right) -1+\mu \left( G\left[ V\left( G\right) -N_{G}\left[ A\right] \right] \right) =\mu \left( G\right) -1. \end{aligned}$$

Thus, v is a \(\mu \)-critical vertex. \(\square \)

Let us consider the graphs from Figure 8. The set \(\left\{ a_{1},a_{4}\right\} \in \textrm{Ind}(G_{1})\) and there is a matching from \(N\left( \left\{ a_{1},a_{4}\right\} \right) \) into \(\left\{ a_{1},a_{4}\right\} \); consequently, this matching is included in a maximum matching of \(G_{1}\). On the other hand, the set \(\left\{ v\right\} \in \textrm{Ind}(G_{2})\) and there no a matching from \(N\left( \left\{ v\right\} \right) \) into \(\left\{ v\right\} \); however, every matching of \(G_{2}\) can be enlarged to a maximum matching. This fact shows that the converse of Theorem 3.4 is not generally true.

Fig. 8
figure 8

\(\mu \left( G_{1}\right) =3\) and \(\mu \left( G_{2}\right) =2\)

Full size image

Theorem 3.4 and Theorem 2.6(iii) immediately imply the following.

Corollary 3.5

If A is a critical independent set in a graph G, then every matching from \(N_{G}(A)\) into A can be enlarged to a maximum matching of G, and every vertex of \(N_{G}(A)\) is \(\mu \)-critical.

Proposition 3.6

If A is a critical independent set in G, then \(core \left( G\right) \cap N_{G}\left( A\right) =\emptyset \), and, consequently, \(core \left( G\right) \cap N_{G}\left( diadem \left( G\right) \right) =\emptyset \).

Proof

By Theorem 2.6, there exists a maximum independent set S, such that \(A\subseteq S\). By definition, \(core \left( G\right) \subseteq S\), as well. Hence,

$$ core \left( G\right) \cap N_{G}\left( A\right) \subseteq core \left( G\right) \cap N_{G}\left( S\right) =\emptyset . $$

Thus, \(core \left( G\right) \cap N_{G}\left( A\right) =\emptyset \).

Finally, we obtain

$$\begin{aligned} core \left( G\right) \cap N_{G}\left( diadem \left( G\right) \right)&=core \left( G\right) \cap N\left( {\displaystyle \bigcup \limits _{A\in Crit(G)}} A\right) =\\ core \left( G\right) \cap {\displaystyle \bigcup \limits _{A\in Crit(G)}} N\left( A\right)&= {\displaystyle \bigcup \limits _{A\in Crit(G)}} \left( core \left( G\right) \cap N\left( A\right) \right) =\emptyset , \end{aligned}$$

as claimed. \(\square \)

Example 3.7

Consider the 1-König-Egerváry graphs from Figure 9: \(n\left( G_{1}\right) =n\left( G_{2}\right) =8\), \(d\left( G_{1}\right) =d\left( G_{2}\right) =1\), \(\xi \left( G_{1}\right) =\xi \left( G_{2}\right) =2\), \(\alpha ^{\prime }\left( G_{1}\right) =\alpha ^{\prime }\left( G_{2}\right) =3\), while \(\varrho \left( G_{1}\right) =4\) and \(\varrho \left( G_{2}\right) =3\).

Fig. 9
figure 9

1-König-Egerváry graphs with \(\varrho (G_{1})=4\) and \(\varrho (G_{2})=3\)

Full size image

Theorem 3.8

If G is a 1-König-Egerváry graph, then

$$ \varrho _{v}\left( G\right) \le n\left( G\right) +d\left( G\right) -\xi \left( G\right) -\beta (G). $$

Proof

Clearly, \(diadem \left( G\right) = {\displaystyle \bigcup \limits _{A\in Crit(G)}} A= {\displaystyle \bigcup \limits _{A\in MaxCrit(G)}} A\), since every critical set is a subset of a maximum critical set (Theorem 2.6(ii)). Hence, we infer that

$$ N_{G}\left( diadem \left( G\right) \right) =N\left( {\displaystyle \bigcup \limits _{A\in MaxCrit(G)}} A\right) = {\displaystyle \bigcup \limits _{A\in MaxCrit(G)}} N_{G}\left( A\right) . $$

Then, Theorem 3.3, Theorem 3.4 and Proposition 3.6 imply

$$\begin{aligned} \varrho _{v}\left( G\right) \le n\left( G\right) -\xi \left( G\right) -\left| {\displaystyle \bigcup \limits _{A\in MaxCrit(G)}} N_{G}\left( A\right) \right| =n\left( G\right) -\xi \left( G\right) -\left| N_{G}\left( diadem \left( G\right) \right) \right| . \end{aligned}$$

By Theorem 1.3, \(diadem \left( G\right) \) is critical in G, as union of critical sets. Hence, \(d\left( G\right) =\left| diadem \left( G\right) \right| -\left| N_{G}\left( diadem \left( G\right) \right) \right| \). Thus,

$$ \varrho _{v}\left( G\right) \le n\left( G\right) +d\left( G\right) -\xi \left( G\right) -\beta (G), $$

and this completes the proof. \(\square \)

It is worth mentioning that the 1-König-Egerváry graphs in Figure 10 point out to the fact that Theorem 3.8 presents a tight inequality:

  • \(n\left( G_{1}\right) =9\), \(\alpha \left( G_{1}\right) =\mu \left( G_{1}\right) =4\), \(d\left( G_{1}\right) =\xi \left( G_{1}\right) =0\), \(diadem \left( G_{1}\right) =\left\{ u,v,w\right\} \), \(\varrho _{v}\left( G_{1}\right) =6=n\left( G_{1}\right) +d\left( G_{1}\right) -\xi \left( G_{1}\right) -\beta \left( G_{1}\right) \);

  • \(n\left( G_{2}\right) =9\), \(\alpha \left( G_{2}\right) =\mu \left( G_{2}\right) =4\), \(d\left( G_{2}\right) =0\), \(\xi \left( G_{2}\right) =1\), \(diadem \left( G_{2}\right) =\left\{ a,b\right\} \), \(\varrho _{v}\left( G_{2}\right) =5<n\left( G_{2}\right) +d\left( G_{2}\right) -\xi \left( G_{2}\right) -\beta \left( G_{2}\right) \).

Fig. 10
figure 10

Tight inequality examples

Full size image

Theorem 3.8 and the definition of \(\alpha ^{\prime }\left( G\right) \) imply the following.

Corollary 3.9

If G is 1-König-Egerváry graph, then

$$ \varrho \left( G\right) \le n\left( G\right) +d\left( G\right) -\xi \left( G\right) -\alpha ^{\prime }\left( G\right) . $$

Remark 3.10

Example 3.7 shows that the inequality presented in Theorem 3.8 is stronger than its counterpart presented in Corollary 3.9. To see it, pay attention to the fact that for the graph \(G_{2}\) in Figure 9 the set \(\left\{ u,v,y\right\} \) is maximum critical independent as well as \(\left\{ u,v,x\right\} \), which is different in comparison with the graph \(G_{1}\) in the same figure, where there is only one maximum critical independent set \(\left\{ a,b,c\right\} \).

If G is a 1-König-Egerváry graph (a general graph - \(K_{2q}\)), then \(N( diadem \left( G\right) ) \) may be a proper subset of the set of all \(\mu \)-critical vertices of G that are not \(\alpha \)-critical. In other words, it is not true that a vertex v is \(\mu \)-critical and not \(\alpha \)-critical if and only if there exists a (maximum) critical independent set A such that \(v\in N_{G}\left( A\right) \).

4 An application

A graph G is a critical vertex almost König-Egerváry graph, if G is not a König-Egerváry graph, but \(G-v\) is a König-Egerváry graph for every \(v\in V(G)\), i.e., \(\varrho _{v}\left( G\right) =n\left( G\right) \). For instance, every \(C_{2k+1}\) is a critical vertex almost König-Egerváry graph. It is worth reminding that, by Theorem 2.6, G is 1-König-Egerváry.

Theorem 4.1

The graph G is critical vertex almost König-Egerváry if and only if G is 1-König-Egerváry without perfect matchings, \(\xi \left( G\right) =0\) and \(\beta \left( G\right) =0\).

Proof

“If” Part. Suppose that G is critical vertex almost König-Egerváry. By Theorem 2.6, G is a 1-König-Egerváry graph. For every \(v\in V\left( G\right) \) we have

$$ \alpha \left( G\right) +\mu \left( G\right) =n\left( G\right) -1=\alpha \left( G-v\right) +\mu \left( G-v\right) \text {,} $$

which, by Lemma 1.7, means that \(\alpha \left( G\right) =\alpha \left( G-v\right) \) and \(\mu \left( G\right) =\mu \left( G-v\right) \). In other words, G has neither \(\alpha \)-critical vertices (\(core \left( G\right) =\emptyset \)) nor perfect matchings. In may be put down like \(\xi \left( G\right) =0\) and \(\mu \left( G\right) <\frac{n\left( G\right) }{2}\).

By Theorem 2.6 and Theorem 3.8, we know that

$$ \varrho _{v}\left( G\right) \le n\left( G\right) +d\left( G\right) -\xi \left( G\right) -\beta (G). $$

Hence,

$$\begin{aligned} 0&\le d\left( G\right) -\beta (G)\Longleftrightarrow \\ \left| diadem \left( G\right) \right|&\le \left| diadem \left( G\right) \right| -\left| N\left( diadem \left( G\right) \right) \right| \Longleftrightarrow \\ \left| N\left( diadem \left( G\right) \right) \right|&=0\Longleftrightarrow \left| diadem \left( G\right) \right| =0\Longleftrightarrow diadem \left( G\right) =\emptyset , \end{aligned}$$

which means \(\beta \left( G\right) =0\) and, consequently, completes the proof.

“Only If” Part. Suppose that G is a 1-König-Egerváry graph (i.e., \(\alpha \left( G\right) +\mu \left( G\right) =n\left( G\right) -1\)) without perfect matchings (i.e., \(\mu \left( G\right) <\frac{n\left( G\right) }{2}\)), \(\xi \left( G\right) =0\) and \(\beta \left( G\right) =0\). We have to prove that \(\alpha \left( G-v\right) +\mu \left( G-v\right) =n\left( G-v\right) \) for each \(v\in V\left( G\right) \).

Since \(\xi \left( G\right) =0\), there are no \(\alpha \)-critical vertices, i.e., \(\alpha \left( G\right) =\alpha \left( G-v\right) \) for every \(v\in V\left( G\right) \). Hence, it is enough to prove that \(\mu \left( G\right) =\mu \left( G-v\right) \) for every \(v\in V\left( G\right) \).

The condition that \(\beta \left( G\right) =0\) is equivalent to the fact that G has only one critical set, namely, the empty set. In other words, G is regularizable, i.e., \(\left| A\right| <\left| N_{G}\left( A\right) \right| \) for every non-empty independent set A of G [2]. In particular, if \(S\in \Omega \left( G\right) \), then

$$ \left| S\right|<\left| N_{G}\left( S\right) \right| =\left| V\right| -\left| S\right| \Longleftrightarrow \alpha \left( G\right) <\frac{n\left( G\right) }{2}. $$

Claim 1. \(\alpha \left( G\right) =\mu \left( G\right) \).

Theorem 2.5(i),(ii) imply \(\mu (G)\le \alpha (G)\), because G has no perfect matchings. On the other hand, if \(S\in \Omega \left( G\right) \), then by Hall’s Theorem, there is a matching from S into \(V\left( G\right) -S\), because G is regularizable. Therefore, \(\alpha (G)\le \mu (G)\). Thus \(\alpha \left( G\right) =\mu \left( G\right) \).

Claim 2. \(\mu \left( G\right) =\mu \left( G-v\right) \) for each \(v\in V\left( G\right) \).

Let \(v\in V\left( G\right) \). Hence, there exists a maximum independent set S, such that \(v\notin S\), because \(core \left( G\right) =\emptyset \). As mentioned above, \(\left| A\right| <\left| N_{G}\left( A\right) \right| \) for every \(A\in Ind \left( G\right) \). Therefore, \(\left| A\right| \le \left| N_{G-v}\left( A\right) \right| \) for every independent set \(A\subseteq S\). Consequently, by Hall’s Theorem, there is a matching, say M, from S into \(V\left( G\right) -\left\{ v\right\} -S\). Clearly, by Lemma 1.7(iii) and Claim 1,

$$ \mu \left( G\right) \ge \mu \left( G-v\right) \ge \left| M\right| =\left| S\right| =\alpha \left( G\right) =\mu \left( G\right) . $$

Finally, \(\mu \left( G-v\right) =\mu \left( G\right) \), which completes the whole proof. \(\square \)

For example, the Friendship graph \(F_{k}\) is a 1-König-Egerváry graph, \(core \left( F_{k}\right) =\emptyset \), \(diadem \left( G\right) =\emptyset \), and, by Theorem 4.1, \(F_{k}\) is critical vertex almost König-Egerváry.

Remark 4.2

Claim 1 of Theorem 4.1 mentions that if G is critical vertex almost König-Egerváry, then \(\alpha \left( G\right) =\mu \left( G\right) \). Since G is 1-König-Egerváry, it follows that

$$ n\left( G\right) =\alpha \left( G\right) +\mu \left( G\right) +1=2\alpha \left( G\right) +1, $$

i.e., the order of G is odd.

The following examples point out to the fact that conditions of Theorem 4.1 can not be weakened.

  • There are 1-König-Egerváry graphs with a perfect matching having \(diadem \left( G\right) =\emptyset \) and \(core \left( G\right) =\emptyset \), that are not critical vertex almost König-Egerváry, for instance \(K_{4}\).

  • The graph \(K_{5}\) shows that being a graph without perfect matchings, \(core \left( G\right) =\emptyset \), \(diadem \left( G\right) =\emptyset \) is not enough to be critical vertex almost König-Egerváry.

  • The 1-König-Egerváry graph \(G_{2}\) in Figure 4 has \(\alpha \left( G_{2}\right) =\mu \left( G_{2}\right) \), and \(diadem \left( G\right) =\emptyset \), while \(core \left( G\right) \ne \emptyset \), and, consequently, \(G_{2}\) is not critical vertex almost König-Egerváry.

  • The 1-König-Egerváry graph \(G_{2}\) in Figure 5 has \(\alpha \left( G_{2}\right) =\mu \left( G_{2}\right) \), and \(core \left( G\right) =\emptyset \), while \(diadem \left( G\right) \ne \emptyset \), and, consequently, \(G_{2}\) is not critical vertex almost König-Egerváry.

Using Theorem 4.1 and the corresponding theorem from [22] , we may conclude with the following.

Theorem 4.3

Let \(G\ \)be a graph of order \(n\left( G\right) \ge 1\). Then \(\varrho _{v}\left( G\right) =n\left( G\right) \) if and only if either

(i) G is König-Egerváry and \(core \left( G\right) =\ker (G)\);

or

(ii) G is 1-König-Egerváry without perfect matchings, \(\xi \left( G\right) =0\) and \(\beta \left( G\right) =0\).

It is worth mentioning that if a graph G is neither König-Egerváry nor 1-König-Egerváry, then \(\varrho _{v}\left( G\right) =0\) in accordance with Theorem 3.1.

5 Conclusions

Theorem 5.1

If there is an edge \(e\in E\left( G\right) \), such that \(G-e\) is a König-Egerváry graph, then G is either König-Egerváry or 1-König-Egerváry.

Proof

By definition, we know that

$$ \alpha (G-e)+\mu (G-e)=n\left( G-e\right) =n\left( G\right) . $$

Case 1. \(\alpha (G-e)=\alpha (G)\).

Thus \(\alpha (G)+\mu (G-e)=n\left( G\right) \). If \(\mu (G-e)=\mu (G)\), then \(\alpha (G)+\mu (G)=n\left( G\right) \), which means that G is a König-Egerváry graph; otherwise, \(\mu (G-e)=\mu (G)-1\) and, consequently, \(\alpha (G)+\mu (G)=n\left( G\right) -1\), i.e., G is a 1-König-Egerváry graph.

Case 2. \(\alpha (G-e)=\alpha (G)+1\).

Thus \(\alpha (G)+1+\mu (G-e)=n\left( G\right) \). If \(\mu (G-e)=\mu (G)\), then \(\alpha (G)+\mu (G)=n\left( G\right) -1\), which means that G is a 1-König-Egerváry graph; otherwise, \(\mu (G-e)=\mu (G)-1\) and, consequently, \(\alpha (G)+\mu (G)=n\left( G\right) \), i.e., G is a König-Egerváry graph. \(\square \)

By Theorem 5.1, if G is neither König-Egerváry nor 1-König-Egerváry, then \(\varrho _{e}\left( G\right) =0\). Theorem 1.10 claims that if G is König-Egerváry, then \(\varrho _{e}\left( G\right) \le m\left( G\right) -\xi \left( G\right) +\varepsilon \left( G\right) \). It justifies the following.

Problem 5.2

Bound \(\varrho _{e}\left( G\right) \) using various graph invariants for 1-König- Egerváry graphs.

A graph G is a critical edge almost König-Egerváry graph, if G is not a König-Egerváry graph, but \(G-e\) is a König-Egerváry graph for every \(e\in E(G)\), i.e., \(\varrho _{e}\left( G\right) =m\left( G\right) \). For instance, every \(C_{2k+1}\) is a critical edge almost König-Egerváry graph. It is worth reminding that, by Theorem 2.6, G is 1-König-Egerváry.

Lemma 5.3

If G is a critical edge almost König-Egerváry graph, then every edge is \(\alpha \)-critical.

Proof

For each \(e\in E\left( G\right) \), we have

$$ \alpha \left( G\right) +\mu \left( G\right) \le \alpha \left( G-e\right) +\mu \left( G\right) \le n\left( G\right) =\alpha \left( G-e\right) +\mu \left( G-e\right) \text {,} $$

which implies that \(\mu \left( G\right) \le \mu \left( G-e\right) \). Therefore, \(\mu \left( G-e\right) =\mu \left( G\right) \) and further,

$$ \alpha \left( G\right) +\mu \left( G\right) <n\left( G\right) =\alpha \left( G-e\right) +\mu \left( G\right) $$

ensures that \(e\in E(G)\) must be \(\alpha \)-critical. \(\square \)

The converse is not true; e.g., \(K_{n},n\ge 5\).

Problem 5.4

Characterize critical edge almost König-Egerváry graphs.

Theorem 1.10 claims that if G is König-Egerváry, then \(\varrho _{v}\left( G\right) =n\left( G\right) -\xi \left( G\right) +\varepsilon \left( G\right) \). It justifies the following.

Problem 5.5

Express \(\varrho _{v}\left( G\right) \) using various graph invariants for 1-König- Egerváry graphs.