1. Introduction
Let
be a polynomial ring over a field
K with the standard grading, and let
I be a graded ideal of
S. A prime ideal
of
S is an
associated prime of
if
for some
, where
is the set of all
such that
. The set of associated primes of
is denoted by
, and the set of maximal elements of
with respect to inclusion is denoted by
. The v-
number of
I, denoted
, is the following invariant of
I that was introduced in [
1] to study the asymptotic behavior of the minimum distance of projective Reed–Muller-type codes, Corollary 4.7 in [
1]:
One can define the v-number of
I locally at each associated prime
of
I:
For a graded module
, we define
. By convention, we set
. Part (d) of the next result was shown in Proposition 4.2 in [
1] for unmixed graded ideals. The next result gives a formula for the v-number of any graded ideal.
Theorem 1. Let be a graded ideal, and let . The following hold:
- (a)
If is a homogeneous minimal generating set of , then: - (b)
;
- (c)
with equality if ;
- (d)
If I has no embedded primes, then
The formulas of Parts (a) and (b) give an algorithm to compute the v-number using
Macaulay2 [
2] (Example 1, Procedure A1 in
Appendix A).
The v-number of nongraded ideals was used in [
3] to compute the regularity index of the minimum distance function of affine Reed–Muller-type codes, Proposition 6.2 in [
3]. In this case, one considers the vanishing ideal of a set of affine points over a finite field.
For certain classes of graded ideals,
is a lower bound for
, the regularity of the quotient ring
(Definition 1); see [
1,
4,
5]. There are examples of ideals where
[
4]. It is an open problem whether
holds for any squarefree monomial ideal. Upper and lower bounds for the regularity of edge ideals and their powers were given in [
6,
7,
8,
9,
10,
11,
12,
13,
14,
15]; see
Section 2. Using the polarization technique of Fröberg [
16], we give an upper bound for the regularity of a monomial ideal
I in terms of the dimension of
and the exponents of the monomials that generate
I (Proposition 2).
Let
G be a graph with vertex set
and edge set
. If
, we can regard each vertex
as a variable of the polynomial ring
and think of each edge
of
G as the quadratic monomial
of
S. The
edge ideal of
G is the squarefree monomial ideal of
S, defined as:
This ideal, introduced in [
17], has been studied in the literature from different perspectives; see [
18,
19,
20,
21,
22,
23,
24,
25,
26] and the references therein. We use induced matchings of
G to compare the v-number of
with the regularity of
for certain families of graphs.
A subset
C of
is a
vertex cover of
G if every edge of
G is incident with at least one vertex in
C. A vertex cover
C of
G is
minimal if each proper subset of
C is not a vertex cover of
G. A subset
A of
is called
stable if no two points in
A are joined by an edge. Note that a set of vertices
A is a (maximal) stable set of
G if and only if
is a (minimal) vertex cover of
G. The
stability number of
G, denoted by
, is the cardinality of a maximum stable set of
G, and the
covering number of
G, denoted
, is the cardinality of a minimum vertex cover of
G. We introduce the following two families of stable sets:
According to Theorem 3.5 in [
4],
and the
-number of
is given by:
The v-number of is a combinatorial invariant of G that has been used to characterize the family of -graphs (see the discussion below after Corollary 1). We can define the v-number of a graph G as and study from the viewpoint of graph theory.
A set P of pairwise disjoint edges of G is called a matching. A matching is perfect if . An induced matching of a graph G is a matching of G such that the only edges of G contained in are . The matching number of G, denoted , is the maximum cardinality of a matching of G, and the induced matching number of G, denoted , is the number of edges in the largest induced matching.
The graph
G is
well-covered if every maximal stable set of
G is of the same size, and
G is
very well-covered if
G is well-covered, has no isolated vertices, and
. The class of very well-covered graphs includes the bipartite well-covered graphs without isolated vertices [
27,
28] and the whisker graphs [
24] (p. 392) (Lemma 1). A graph without isolated vertices is very well-covered if and only if
G is well-covered and
(Proposition 1). One of the properties of very well-covered graphs that will be used to show the following theorem is that they can be classified using combinatorial properties of a perfect matching, as was shown by Favaron, Theorem 1.2 in [
29] (Theorem 7, cf. Theorem 6).
We come to one of our main results.
Theorem 2. Let G be a very well-covered graph, and let be a perfect matching of G. Then, there is an induced submatching of P and such that and for each . Furthermore, .
Let
G be a graph, and let
be its whisker graph (
Section 2). As a consequence, we recover a result of [
4] showing that the v-number of
is bounded from above by the regularity of the quotient ring
(Corollary 3). The independent domination number of
G, denoted by
, is the minimum size of a maximal stable set, Proposition 2 in [
30]:
and
is equal to the v-number of the whisker graph
of
G, Theorem 3.19(a) in [
4].
A cycle of length s is denoted by . The inequality of Theorem 2 is false if we only assume that G is a well-covered graph, since the cycle is a well-covered graph, but one has . We prove that is the only cycle where the inequality fails.
Theorem 3. Let be an s-cycle, and let be its edge ideal. Then, if and only if .
If , we denote the closed neighborhood of v by . A vertex v of G is called simplicial if the induced subgraph on the vertex set is a complete graph. A subgraph H of G is called a simplex if for some simplicial vertex v. A graph G is simplicial if every vertex of G is either simplicial or is adjacent to a simplicial vertex of G.
If
A is a stable set of a graph
G,
is a complete subgraph of
G for
, and
is a partition of
, then
, Theorem 2 in [
15]. We consider a special type of partition of
that allows us to link
with induced matchings of
G. A graph
G has a
simplicial partition if
G has simplexes
, such that
is a partition of
. Our next result shows that
if
G has a simplicial partition.
Theorem 4. Let G be a graph with simplexes , such that is a partition of . If G has no isolated vertices, then there is , and there are simplicial vertices of G and integers such that is an induced matching of G and is the induced subgraph on for . Furthermore, .
As a consequence, using a result of Finbow, Hartnell, and Nowakowski that classifies the connected well-covered graphs without four and five cycles, Theorem 1.1 in [
31] (Theorem 8), we show other families of graphs where the induced matching number of
G is an upper bound for the v-number of
.
Corollary 1. Let G be a well-covered graph, and let be its edge ideal. If G is simplicial or G is connected and contains neither four, nor five cycles, then: A vertex v of a graph G is called a shedding vertex if each stable set of is not a maximal stable set of . We prove that every vertex of G is a shedding vertex if and only if (Proposition 4).
A graph
G belongs to class
if
and any two disjoint stable sets
are contained in two disjoint maximum stable sets
with
for
. A graph
G is in
if and only if
G is well-covered,
is well-covered for all
, and
G has no isolated vertices, Theorem 2.2 in [
32]. A graph
G without isolated vertices is in
if and only if
, Theorem 4.5 in [
4]. As an application we recover the only if implication of this result (Corollary 5). Using the fact that a graph
G without isolated vertices is in
if and only if
G is well-covered and
, Theorem 4.3 in [
4], by Proposition 4, we recover the fact that a graph
G without isolated vertices is in
if and only if
G is well-covered and every
is a shedding vertex, Theorem 3.9 in [
32]. For other characterizations of graphs in
, see [
32,
33] and the references therein.
In
Section 5, we show examples illustrating some of our results. In particular, in Example 3, we compute the combinatorial and algebraic invariants of the well-covered graphs
and
that are depicted in
Figure 1. These two graphs occur in the classification of connected well-covered graphs without four and five cycles, Theorem 1.1 in [
31] (Theorem 8). A related result is the characterization of well-covered graphs of girth at least five given in [
34].
For all unexplained terminology and additional information, we refer to [
35,
36] for the theory of graphs and [
19,
21,
25] for the theory of edge ideals and monomial ideals.
2. Preliminaries
In this section, we give some definitions and present some well-known results that will be used in the following sections. To avoid repetition, we continue to employ the notations and definitions used in
Section 1.
Definition 1 ([
37]).
Let be a graded ideal, and let be the minimal graded free resolution of as an S-module:The Castelnuovo–Mumford regularity of (regularity of ) is defined as: The integer g, denoted , is the projective dimension of .
Let G be a graph with vertex set . Given , the induced subgraph on A, denoted , is the maximal subgraph of G with vertex set A. The edges of are all the edges of G that are contained in A. The induced subgraph of G on the vertex set is denoted by . If v is a vertex of G, then we denote the neighborhood of v by and the closed neighborhood of v by . Recall that is the set of all vertices of G that are adjacent to v. If , we set .
Theorem 5 ([
38]).
If a graph G is well-covered and is not complete, then is well-covered for all v in . Moreover, . If G is a graph, then . We say that G is a König graph if . This notion can be used to classify very well-covered graphs (Proposition 1).
Theorem 6 ([
39], Theorem 5, and [
40], Lemma 2.3).
Let G be a graph without isolated vertices. If G is a graph without 3, 5, and 7 cycles or G is a König graph, then G is well-covered if and only if G is very well-covered. Definition 2. A perfect matching P of a graph G is said to have Property (P) if for all , , and , one has .
Remark 1. Let P be a perfect matching of a graph G with Property(P). Note that if and , then and cannot be both in because G has no loops. In other words, G has no triangle containing an edge in P.
Theorem 7 ([
29], Theorem 1.2).
The following conditions are equivalent for a graph G:- 1.
G is very well-covered;
- 2.
G has a perfect matching with Property (P);
- 3.
G has a perfect matching, and each perfect matching of G has Property (P).
Let
G be a graph with vertex set
, and let
be a new set of vertices. The
whisker graph or
suspension of
G, denoted by
, is the graph obtained from
G by attaching to each vertex
a new vertex
and a new edge
. The edge
is called a
whisker or
pendant edge. The graph
was introduced in [
24] as a device to study the numerical invariants and properties of graphs and edge ideals.
Lemma 1. Let G be a graph without isolated vertices. The following hold:
- (a)
If G is a bipartite well-covered graph, then G is very well-covered;
- (b)
The whisker graph of G is very well-covered.
Proof. (a) A bipartite well-covered graph without isolated vertices has a perfect matching
P that satisfies Property
(P), Theorem 1.1 in [
28]. Thus, by Theorem 7,
G is very well-covered;
(b) The perfect matching of the whisker graph satisfies Property (P) and, by Theorem 7, G is very well-covered. □
Proposition 1 ([
41], Lemma 17).
Let G be a graph without isolated vertices. Then, G is a very well-covered graph if and only if G is well-covered and . Proof. ⇒) Assume that G is very well-covered. Then, . It suffices to show that . In general, . By Theorem 7, G has a perfect matching . Then, and . Thus, , and one has .
⇐) Assume that G is well-covered and . Let be a matching of G with . We need only to show that . Clearly, is greater than or equal to because . We argue by contradiction assuming that . Pick . As v is not an isolated vertex of G, there is a minimal vertex cover C of G that contains v. As G is well-covered, one has that . Since for and , we obtain , a contradiction. □
We say that a graph G is in the family if there exists where for each i, is simplicial, , and is a partition of .
Theorem 8 ([
31], Theorem 1.1).
Let G be a connected graph that contains neither four, nor five cycles, and let and be the two graphs in Figure 1. Then, G is a well-covered graph if and only if or . Theorem 9. Let G be a graph. The following hold:
- (a)
([
7], Theorem 4.5, [
42])
for all ;- (b)
([
7], Theorem 4.7, [
43])
If G is a forest or G is very well-covered, then:- (c)
([
44], Theorem 1.3)
If G is very well-covered, then .
The projective dimension of the edge ideal of a graph, the Wiener index, the independence polynomial, the h-vector, and the symbolic powers of cover ideals of graphs have been studied for very well-covered graphs [
45,
46,
47,
48,
49,
50,
51].
3. The v-Number of a Graded Ideal
Let
be a polynomial ring over a field
K with the standard grading, and let
I be a graded ideal of
S. In this section, we prove a formula for the v-number of
I that can be used to compute this number using
Macaulay2 [
2]. To avoid repetition, we continue to employ the notations and definitions used in
Section 1 and
Section 2.
Lemma 2. Let be a graded ideal. If for some prime ideal and some , , then , and there is a minimal homogeneous generator of such that and .
Proof. The strict inclusion
is clear because
. Let
be a minimal generating set of
such that
is a homogeneous polynomial for all
i. As
, one has
and
. Then, we can choose homogeneous polynomials
in
S,
p in
I, such that
and
for all
i with
. One has the inclusion
. Indeed, if we take
h in
, then
for all
i and
, thus
. Therefore, using the fact that all
’s are in
, one has the inclusions:
and consequently,
. Hence, by [
25] (p. 74, 2.1.48), we obtain
for some
. As
, we obtain:
Hence, and . □
Theorem 10 (The same as Theorem 1). Let be a graded ideal, and let . The following hold:
- (a)
If is a homogeneous minimal generating set of , then: - (b)
;
- (c)
with equality if ;
- (d)
If I has no embedded primes, then
Proof. (a) Take any homogeneous polynomial
f in
S such that
. Then, by Lemma 2, there is
such that
and
. Thus, the set
is not empty and the inequality:
follows by the definition of
. Now, we can pick a homogeneous polynomial
f in
S such that
and
. Then, by Lemma 2, there is
such that
and
. Thus,
and the inequality “≥” holds;
(b) This follows at once from the definitions of and ;
(c) Pick a homogeneous polynomial g in S such that and . Then, and , that is . Thus, . Now, assume that . To show the reverse inequality, take any homogeneous polynomial f in . Then, and . Since is contained in , there is such that . Hence, and . Thus, and ;
(d) This follows immediately from (b) and (c). □
We give a direct proof of the next result, which in particular relates the v-number of a Cohen–Macaulay monomial ideal to that of , where and .
Corollary 2 ([
4], Proposition 4.9).
Let be a Cohen–Macaulay nonprime graded ideal whose associated primes are generated by linear forms, and let be a regular element on . Then, . Proof. Since the ideal
I has no embedded primes, by Theorem 10d, there are
and
such that
is a minimal generator of
and
. The associated primes of
are contained in
; thus, there is
such that
. Hence,
because
I has no embedded associated primes, and one has the equality
. We claim that
f is not in
. We assume, by contradiction, that
. Then, we can write
, with
a homogeneous polynomial for
,
,
. Hence, one has:
Therefore,
and
, a contradiction because
is a minimal generator of
. This proves that
. Next, we show the equality
. The inclusion “⊂” is clear because
. Take an associated prime
of
. The height of
is equal to
because
is Cohen–Macaulay and the associated primes of
are contained in
. Then:
and consequently,
. Now,
is prime because
is generated by linear forms, and
because
I is Cohen–Macaulay and
h is a regular element on
. Thus,
,
, and
. □
Proposition 2. Let be a monomial ideal minimally generated by , and for each that occurs in a monomial of , let . Then: Proof. To show the inequality, we use the polarization technique due to Fröberg (see [
52] and [
25] (p. 203)). To polarize
I we use the set of new variables:
where
is empty if
. Note that
. A power
of a variable
,
, polarizes to
if
, to
if
, and to
if
. Setting
, the polarization
of
I is the ideal of
generated by
. According to Corollary 1.6.3 in [
21], one has:
As
is squarefree, by Proposition 3.2 in [
4], one has
. Hence, we obtain:
To complete the proof, notice that . □
Given
, where
, the monomial
is denoted by
. A result of Beintema [
53] shows that a zero-dimensional monomial ideal is Gorenstein if and only if it is a complete intersection. (This is also true in dimension one; see Exercise 4.4.19 in [
54].) The next result classifies the complete intersection property using regularity.
Proposition 3. Let I be a monomial ideal of S of dimension zero minimally generated by , where for and for . Then, , with equality if and only if I is a complete intersection.
Proof. The inequality
follows directly from Proposition 2 because
. If
I is a complete intersection, then
, and by Lemma 3.5 in [
55], we obtain
. Conversely, assume that
is equal to
. We argue by contradiction assuming that
. Then, the exponents of the monomial
satisfy
for
because
. The regularity of
is the largest integer
such that
, Proposition 4.14 in [
37]. Pick a monomial
such that
and
. Then,
for
because
is not in
I, and consequently,
for
. Hence,
for some
, a contradiction. □
Remark 2. Note that Proposition 3 follows also from Corollary 3.17 in [56]. Indeed, assume that is equal to . Let be the irreducible decomposition of I, where the ’s are irreducible monomial ideals of S, i.e., ideals generated by powers of variables in S. We argue by contradiction assuming that I is not a complete intersection. Then, I is not irreducible and for all k because for all k. Therefore, by Corollary 3.17 in [56], it follows that because I is -primary, , and , a contradiction. 4. Induced Matchings and the v-Number
In this section, we show that the induced matching number of a graph
G is an upper bound for the v-number of
when
G is very well-covered, or
G has a simplicial partition, or
G is well-covered connected and contains neither four, nor five cycles. We classify when the induced matching number of
G is an upper bound for the v-number of
when
G is a cycle and classify when all vertices of a graph are shedding vertices to gain insight into the family of
-graphs. To avoid repetition, we continue to employ the notations and definitions used in
Section 1 and
Section 2.
Theorem 11 ([
4], Theorem 3.5).
If is the edge ideal of a graph G, then and the -number of I is: Lemma 3. Let A be a stable set of a graph G. If is a vertex cover of G, then .
Proof. We take any
, then there is
such that
. Furthermore,
, since
A is a stable set of
G. Thus,
and consequently,
. Hence,
is not a vertex cover of
G, since
. Therefore,
is a minimal vertex cover of
G and
□
Theorem 12 (The same as Theorem 2). Let G be a very well-covered graph, and let be a perfect matching of G. Then, there is an induced submatching of P and such that and for each . Furthermore, .
Proof. To show the first part, we use induction on . If , we set and , where . Assume . We set , and . By Theorem 7, P satisfies Property (P). Then, satisfies Property (P) as well. Thus, by Theorem 7, is very well-covered with a perfect matching . Hence, by the induction hypothesis, there is an induced submatching of and such that and for each . Consequently, is a minimal vertex cover of . We consider two cases: and :
Case (I). Assume that . Thus, we may assume that there is with . Then, , since P satisfies Property (P). Hence, is a vertex cover of G, since is a vertex cover of and . Therefore, by Lemma 3, , so this case follows by making and ;
Case (II). Assume that
. We set
, then
is a stable set of
and also of
G, since
is an induced matching of
and also of
G. One has the inclusion:
indeed taking
(the case
is similar). If
, then
,
, and
, a contradiction. We claim that
. We assume, by contradiction, that
. Then, there are
such that
. Thus,
, since
P satisfies Property
(P), a contradiction, since
is a stable set of
G. Hence,
, and we may assume:
Next we show that
. If the intersection is nonempty, by Equation (
1), we can pick
z in
, then
and
, a contradiction to Equation (
2). Therefore, by Equation (
1), we obtain the inclusion:
Thus, the edge set
is an induced matching, since
is an induced matching. Setting:
i.e.,
, we obtain
for each
, since
for each
. Note that
is a stable set of
G, since
is a stable set and
. Now, take
. We prove that
. Clearly,
because
. If
, then
. Now, if
, then
for some
y in
, and
. Therefore, we may assume
, then
. Thus, there is
, since
is a vertex cover of
. Then, there is
, such that
. If
, then
. Finally, if
, then by Equation (
3) and the inclusion
, there is
such that
. Therefore,
, since
. This implies,
, since
,
,
, and
P satisfies Property
(P). Thus,
. Hence,
is a vertex cover, and by Lemma 3,
. Therefore, this case follows by making
and
. This completes the induction process.
Next, we show the equality . By the first part, we may assume that , , for , and . Thus, , and since , we obtain . Then, . The inequality follows by Theorem 11, and is clear by the definition of . Finally, the inequality follows directly from Theorem 9. □
Corollary 3 ([
4], Theorem 3.19(b)).
Let G be a graph, and let be its whisker graph. Then: Proof. By Lemma 1, is very well-covered. Thus, by Theorem 12, the v-number of is bounded from above by the regularity of . □
Lemma 4. Let and be integers with . If and , then: Proof. By the division algorithm,
, where
. Then:
Thus, . If , then . This follows using the fact that and . Hence, . Now, assume . We claim that . We assume, by contradiction, that , then . If , then , since , a contradiction, since and . Thus, , and we have . This implies and . Consequently , a contradiction. Therefore, and . □
Theorem 13 (The same as Theorem 3). Let be an s-cycle, and let be its edge ideal. Then, if and only if .
Proof. ⇒) Assume that . If , then and , a contradiction. Thus, .
⇐) Assume that . We can write . The matching , where , is an induced matching of and . Now, we choose a stable set A of , for each one of the following cases:
Case . If , then is a vertex cover of G and ;
Case . If , then is a vertex cover of G and ;
Case . If , then is a vertex cover of G and ;
Case . If , then is a vertex cover of G and .
In each case, and is a vertex cover of G. Therefore, by Lemma 3, . Now, assume , with and an integer. Then, by Lemma 4, if and otherwise. Hence, . Therefore, , since and . □
Remark 3. The induced matching number of the cycle is equal to . The regularity of is equal to , Proposition 10 in [15]. Lemma 5. Let G be a graph without isolated vertices, and let be vertices of G such that is a partition of . If , then:
- (i)
for ;
- (ii)
for .
Proof. (i) Assume that . Clearly, because is a subgraph of G. To show the inclusion “⊃”, take . Then, or . If , then , a contradiction. Thus, , and since is an induced subgraph of G, we obtain or . Thus, ;
(ii) By Part (i), one has
. Then:
Thus, . □
Theorem 14 (The same as Theorem 4). Let G be a graph with simplexes , such that is a partition of . If G has no isolated vertices, then there is , and there are simplicial vertices of G and integers such that is an induced matching of G and is the induced subgraph on for . Furthermore, .
Proof. We proceed by induction on r. If , then , and there is a simplicial vertex of G such that is a complete graph with at least two vertices. Picking , , one has and is an induced matching. Now, assume that . We set . Note that are simplexes of (Lemma 5) and is a partition of . Then, by the induction hypothesis, there is , and there are simplicial vertices of and integers , such that is an induced matching of and for . By Lemma 5, one has for . We can write for some simplicial vertex x of G.
Case (I). Assume that . Then, is a vertex cover of G. Indeed, take any edge e of G. If , then e is an edge of and is covered by . Assume that . If , then there is with . Now, if , then with . This proves that is a vertex cover of G. Hence, by Lemma 3, , and, noticing that is an induced matching of G, this case follows by making and ;
Case (II) Assume that there is such that . Then, is a stable set of G. Furthermore, is a vertex cover of G, since is a vertex cover of , is a complete subgraph of G, and . Thus, by Lemma 3, is in . We set , , and . Then, and is an induced matching of G, since is an induced matching of , and , for . Therefore, this case follows by making and .
The equality is clear. The inequality follows from Theorem 11, and is clear by the definition of . Finally, the inequality follows directly from Theorem 9. □
Corollary 4 (The same as Corollary 1).
Let G be a well-covered graph, and let be its edge ideal. If G is simplicial or G is connected and contains neither four, nor five cycles, then: Proof. Assume that
G is simplicial. Let
be the set of all simplicial vertices of
G. Then,
. As
G is well-covered, by Lemma 2.4 in [
31], for
, either
or
. Thus, there are simplicial vertices
of
G such that
is a partition of
. Setting
for
and applying Theorem 14, we obtain that
. Noticing that
, the inequality
follows from Proposition 2.
Next, assume that G is connected and contains neither four, nor five cycles. Then, by Theorem 8, or . The cases or are treated in Example 3 (cf. Theorem 13). If , then there exists where for each i, is simplicial, , and is a partition of . In particular, G is simplicial, and the asserted inequalities follow from the first part of the proof. □
Proposition 4. Let G be a graph. The following conditions are equivalent:
- 1.
Every vertex of G is a shedding vertex;
- 2.
.
Proof. (1) ⇒ (2) The inclusion follows from Theorem 11. To show the inclusion , we argue by contradiction assuming that there is . Then, D is a stable set of G and is a vertex cover of G. Thus, . Furthermore, since , there is such that is a stable set of G. Then, . However, is a vertex cover of G, then and is a stable set of G. Therefore, and is a stable set of . Now, we prove that is a maximal stable set of . We argue by contradiction assuming that there is , such that is a stable set. Then, , since . Furthermore, , since and , a contradiction, since and is a stable set. Hence, is a maximal stable set of . Therefore, x is not a shedding vertex of G, a contradiction.
(2) ⇒ (1) We assume, by contradiction, that there is such that x is not a shedding vertex. Thus, there is a maximal stable set A of such that . Then, is a minimal vertex cover of and is a stable set of G. Therefore, . Since C is a minimal vertex cover of , we have that for each , there is such that . Consequently, . Furthermore, if , then and . Thus, , since A is a maximal stable set of . Hence, . This implies that is a vertex cover of G, since . Therefore, by Lemma 3, , a contradiction since . □
Lemma 6 ([
32], cf. Corollary 3.3).
If , then every is a shedding vertex. Proof. Let
v be a vertex of
G. We may assume that
G is not a complete graph. Let
A be a stable set of
. We argue by contradiction assuming that
A is a maximal stable set of
. Then, as
G and
are well-covered, we obtain:
According to [
57], Theorem 5, the graph
is in
and
. In particular,
is well-covered and
(cf. Theorem 5). However,
A is a stable set of
and
, a contradiction. □
Corollary 5 ([
4], Theorem 4.5).
If G is a -graph and , then . Proof. By Theorem 11, there is such that . Since G is a -graph, by Lemma 6, every vertex of G is a shedding vertex. Thus, by Proposition 4, , i.e., D is a maximal stable set of G. Furthermore, G is well-covered, since G is a -graph. Hence, . Therefore, . □