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A note on graphs contraction-critical with respect to independence number
Discrete Mathematics 325 (2014) 85–91
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A note on graphs contraction-critical with respect to independence number Michael D. Plummer a,∗ , Akira Saito b a
Department of Mathematics, Vanderbilt University, Nashville, TN 37240, United States
b
Department of Information Science, Nihon University, Sakurajosui 3-25-40, Setagaya-ku, Tokyo 156-8550, Japan
article
info
Article history: Received 1 August 2013 Received in revised form 4 February 2014 Accepted 5 February 2014 Available online 13 March 2014 Keywords: Independent set Contraction-critical Matching 1-extendable Polynomial algorithm
abstract Let α(G) denote the independence number of graph G, i.e., the size of any maximum independent set of vertices. A graph G is contraction-critical (with respect to α ) if α(G/ xy) < α(G), for every edge xy ∈ E (G), where G/ xy denotes the graph obtained from G by shrinking the edge xy to a single vertex xy and deleting the loop thus formed. Let us denote the class of all such contraction-critical graphs by CCR. First it is shown that all graphs in CCR must be bipartite. Let G = (A, B) be a bipartite graph with bipartition A ∪ B. It is then shown that (a) if |A| < |B|, then G ∈ CCR if and only if B is the unique maximum independent set in G if and only if G is 1-expanding, while (b) if |A| = |B|, then G ∈ CCR if and only if A and B are the only two maximum independent sets in G if and only if G is 1-extendable. It follows that CCR graphs can be recognized in polynomial time. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Let α(G) denote the independence number of the graph G, that is, the size of any largest independent set of vertices in G. Problems involving α(G) have proven to be quite difficult on the whole so far. Perhaps this is not so surprising in view of the fact that the determination of α(G) for arbitrary G was one of the first problems to be shown to be NP-complete [10]. On the other hand, for certain special subclasses of graphs, α(G) can be computed in polynomial time. For example, if G is bipartite, then α(G) may be computed in polynomial time by using a polynomial matching algorithm together with two classical results by König and Gallai (cf. [15]). Claw-free graphs constitute a second graph class admitting a polynomial algorithm for determining α (cf. [16,20]). Clearly, if a graph has the property that every maximal independent set is, in fact, maximum (the so-called well-covered graphs), determination of α(G) becomes trivially polynomial, but no polynomial recognition algorithm for well-covered graphs is known. Interesting results about well-covered graphs have been obtained, however. For a sampling, see the surveys [5,17]. Historically, useful information about certain graph parameters has been gleaned from the study of various kinds of ‘‘criticality’’ with respect to the given parameter. For example, graphs which are critical with respect to edge deletion (the so-called α -critical graphs) have been investigated. These are the graphs G for which α(G − e) > α(G), for every edge e ∈ E (G). However, no polynomial recognition algorithm for this graph family is known and these graphs remain far from well-understood. For more on this graph family, see Section 12.1 of [15].
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (M.D. Plummer), [email protected] (A. Saito).
http://dx.doi.org/10.1016/j.disc.2014.02.004 0012-365X/© 2014 Elsevier B.V. All rights reserved.
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The idea of criticality with respect to edge-contraction has proved extremely useful in the study of certain graph properties, notably in the study of graph connectivity (cf. [12]), graph coloring (cf. [9,13,21]) and graph minors (cf. [19]). However, the notion of criticality of edge contraction applied to independence number does not seem to have been investigated heretofore. Let G be a graph and xy an edge in G. The edge xy is be said to be contraction-critical (with respect to α ) if α(G/ xy) < α(G), where G/ xy denotes the graph obtained from G by shrinking the edge xy to a single vertex xy and deleting the loop and any multiple edges thus formed. A graph G is contraction-critical if every edge of G is contraction-critical. Let us denote the class of all such graphs by CCR. (Note that a graph belongs to CCR if and only if all its components belong to CCR.) In the present note, we show, that the CCR graphs turn out to be a well-known polynomially recognizable family in disguise. First we show that all members of the family CCR must be bipartite and then we obtain a characterization theorem. To accomplish this, we shall need the notions of 1-extendability and 1-expandability both of which deal with matchings. Although these two properties may be defined for graphs in general, we will confine both properties to bipartite graphs since the class CCR is bipartite. So let G = (A, B) be a bipartite graph with bipartition A ∪ B where, without loss of generality, we assume that |A| ≤ |B|. Definition ([1]). If G = (A, B) is bipartite with |A| < |B|, then G is 1-expanding if for every edge e of G, there is a matching of all A into B which contains e. An equivalent version of this property is given by the following result. Lemma 1.1 (Lemma 6.2.4 [1]). A connected bipartite graph G = (A, B) with |A| < |B| is 1-expanding if and only if for all X ⊆ A, ∅ ̸= X ̸= A, |N (X )| ≥ |X | + 1. The ‘‘balanced’’ version of the above property is the following. Definition ([6,15]). If G = (A, B) is bipartite with |A| = |B|, then G is 1-extendable if every edge of G lies in a perfect matching in G. This property too admits a useful equivalent statement. Lemma 1.2 ([6,14]). A connected bipartite graph G = (A, B) with |A| = |B| is 1-extendable if and only if for all X ⊆ A, ∅ ̸= X ̸= A, |N (X )| ≥ |X | + 1. In this note we first show that if G ∈ CCR, it must be bipartite. We then obtain the following characterizations: (a) if G = (A, B) is a connected bipartite graph with |A| < |B|, then G ∈ CCR if and only if B is the unique maximum independent set in G if and only if G is 1-expanding, while (b) if G = (A, B) is a connected bipartite graph with |A| = |B|, then G ∈ CCR if and only if A and B are the only two maximum independent sets in G if and only if G is 1-extendable. It follows from these characterizations that, since both the 1-extendable and 1-expanding properties can be checked in polynomial time via matching algorithms, membership in the class CCR is also polynomially verifiable. We shall denote by ν(G) the size of a maximum matching in graph G and by τ (G) the size of a minimum vertex cover in G. Note that we will not consider the single-vertex graph K1 to be bipartite. For all other terminology and notation, see [2]. Note also that in this paper all graphs are finite and simple. 2. Main results Lemma 2.1. If G belongs to CCR, then G is a bipartite graph and every maximum independent set is a partite set of G. In particular, |V (G)| ≤ 2α(G). Proof. Let G be a contraction-critical graph and let I be a maximum independent set in G. If G − I contains an edge e, let G′ = G/ˆe. Then I is an independent set in G′ which implies that α(G′ ) ≥ α(G). This contradicts the definition of a contractioncritical graph. Therefore, V (G) − I is an independent set. Since I is also independent, G is a bipartite graph with bipartition (I , V (G) − I ). Since I is a maximum independent set and V (G) − I is an independent set, |V (G) − I | ≤ |I | = α(G) and hence |V (G)| ≤ 2α(G). The following result is well-known. Lemma 2.2. If G is a connected bipartite graph with bipartition (A, B), then this bipartition is unique.
Corollary 2.3. Suppose G ∈ CCR with bipartition (A, B), where by Lemma 2.1 we may assume |B| = α(G). Then if |A| < |B|, G contains exactly one maximum independent set, namely B, and if |A| = |B|, G contains exactly two maximum independent sets, namely A and B.
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Fig. 2.1. The edge e is not contraction-critical.
Fig. 2.2. The edge e is not contraction-critical.
Proof. Let I be a maximum independent set in G. Then (I , V (G) − I ) is a bipartition of G by Lemma 2.1 and by Lemma 2.2, this bipartition is unique. The result follows. Remark. A bipartite graph can be connected and have exactly two maximum independent sets, but fail to be contractioncritical. An example is given in Fig. 2.1 where α = 5 and the edge e is not contraction-critical. Lemma 2.4. Suppose that G is a connected bipartite graph with bipartition (A, B) and suppose G ̸= K2 . (a) If |A| < |B| and B is the only maximum independent set in G, then G ∈ CCR. (b) If |A| = |B| and A and B are the only maximum independent sets in G, then G ∈ CCR. Proof. (a) Suppose G is a connected bipartite graph with |A| < |B| and that B is the only maximum independent set in G, ) = α(G). but suppose G ̸∈ CCR. So there is an a ∈ A and a b ∈ B with a adjacent to b such that α(G/ab ˆ ˆ Let I be a maximum independent set in G/ab. Note that |I | = α(G/ab) = α(G). If ab ̸∈ ˆI, then ˆI is an independent set in G and hence a maximum independent set in G. But this is a contradiction since b ∈ B − ˆI and hence B ̸= ˆI. Thus we have ∈ ˆI. Let Ia = (ˆI − {ab }) ∪ {a}. Then Ia is a maximum independent set in G, since |I | = |ˆI | = α(G). But this is again a ab contradiction since a ∈ Ia − B and hence Ia ̸= B. (b) If |A| = 1, then since |B| = |A| and G is connected, we have that G = K2 . This contradicts the assumption. Hence we may suppose both |A| ≥ 2 and |B| ≥ 2. Again, as in Case (a), suppose G ̸∈ CCR. So there exist vertices a ∈ A and b ∈ B such that a and b are adjacent, but ) = α(G). α(G/ab
lies in every maximum independent set in G/ab. Claim 1. Vertex ab but ab ̸∈ ˆI. Now ˆI ⊆ V (G) and |ˆI | = α(G/ab ) = α(G) Suppose, to the contrary, that ˆI is a maximum independent set in G/ab, and so ˆI is a maximum independent set in G. But then either ˆI = A or ˆI = B. Without loss of generality, suppose ˆI = B. But b ∈ B and b ̸∈ ˆI, which is a contradiction. This proves Claim 1. then ˆI ∩ A ̸= ∅ and ˆI ∩ B ̸= ∅. Claim 2. If ˆI is a maximum independent set in G/ab, }) ∪ {b} = I ′ ⊆ B. By symmetry, it is enough to show that ˆI ∩ A ̸= ∅. Suppose, to the contrary, that ˆI ∩ A = ∅. Hence (ˆI − {ab But |I ′ | = α(G), so I ′ = B. (I ′ ̸= A, since b ∈ I ′ .) But |B| ≥ 2, so I ′ − {b} ̸= ∅. So I ′′ = (I ′ − {b}) ∪ {a} is independent in G and hence |I ′′ | = α(G). So |I ′′ | = |I ′ | = α(G) and I ′′ ∩ B ̸= ∅ ̸= I ′′ ∩ A. But then I ′′ is a maximum independent set in G, but A ̸= I ′′ ̸= B, which is a contradiction. This completes the proof of Claim 2. and hence ab ∈ ˆI. Let I0 = (ˆI −{ab })∪{a}. Then I0 is independent Again, suppose ˆI is a maximum independent set in G/ab in G and |I0 | = |ˆI | = α(G) and so I0 is a maximum independent set in G. But I0 ∩ A ̸= ∅ ̸= I0 ∩ B by Claim 2. So I0 ̸= A and I0 ̸= B, which is a contradiction. Let us point out, however, that if G = (A, B) is a connected bipartite graph where |A| ≤ |B| and if G has a unique maximum independent set I, it is not necessarily true that I = B nor is it necessarily true that G ∈ CCR. (See the bipartite graph in Fig. 2.2 in which I = {a2 , a3 , b2 , b3 , b4 } is the unique maximum independent set, I ̸= B and edge e = a1 b1 is not contraction-critical.)
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However, the next remark is clear. Remark. Every graph has a maximum independent set that contains all of its degree 1 vertices. In particular, if a graph G has a unique maximum independent set, it contains all its degree 1 vertices. Theorem 2.5. Suppose G = (A, B) is a connected bipartite graph. (a) If |A| = |B| and G is 1-extendable, then A and B are the unique maximum independent sets in G. (b) If |A| < |B| and G is 1-expanding, then B is the unique maximum independent set in G. Proof. Let I be a maximum independent set in G and let IA = I ∩ A and IB = I ∩ B. If IA ̸= ∅, and IA ̸= A, let J = IB ∪ N (IA ). Then J ⊆ B and hence J is independent. Moreover, since I is independent, IB ∩ N (IA ) = ∅, and since G is a 1-extendable graph or a 1-expanding graph with |A| < |B|, by Lemmas 1.1 and 1.2 it follows that |N (IA )| ≥ |IA | + 1. However, this implies |J | = |IB | + |N (IA )| ≥ |IA | + |IB | + 1 = |I | + 1, which contradicts the maximality of I. Therefore, we have IA = ∅ or IA = A. If IA = A, then IB = ∅ since G is connected, and hence I = A. If IA = ∅, then I ⊆ B, and by the maximality of I, we have I = B. Therefore, we have either I = A or I = B, which proves (a). If |A| < |B|, then A cannot be a maximum independent set, and I = B, which proves (b). The next corollary is immediate by Theorem 2.5 and Lemma 2.4. Corollary 2.6. If G = (A, B) is a 1-expanding or 1-extendable connected bipartite graph of order at least 3, then G ∈ CCR.
Now we assemble the above results into a characterization theorem for graphs in CCR. Theorem 2.7. Let G be a connected graph, G ̸= K2 . Then the following are equivalent: (a) G ∈ CCR. (b) G is bipartite with bipartition (A, B) where we may assume |A| ≤ |B| and if |A| < |B|, B is the only maximum independent set in G, whereas if |A| = |B|, A and B are the only maximum independent sets in G. (c) G is bipartite with bipartition (A, B) where we may assume |A| ≤ |B| and if |A| < |B|, G is 1-expanding whereas if |A| = |B|, G is 1-extendable. Proof. Suppose G ∈ CCR. Then by Lemma 2.1, G is bipartite and if (A, B) is the bipartition of G, we may assume B is a maximum independent set in G. Then by Corollary 2.3, part (b) follows. Suppose next that part (b) holds. We prove part (c) in the case when |A| < |B|; the proof when |A| = |B| is quite similar. Suppose, to the contrary, that G is not 1-expanding. Then there is a subset X of A, ∅ ̸= X ̸= A, such that |N (X )| ≤ |X | by Lemma 1.1. But then the set S = (B − N (X )) ∪ X is independent and |S | ≥ |B|. Hence |S | = |B| and S is a maximum independent set. But S ̸= B, which is a contradiction. So part (c) holds. Suppose now that part (c) holds. Then part (b) holds by Theorem 2.5. Finally if part (b) holds, then part (a) holds by Lemma 2.4. Remark. The assumption that G ̸= K2 is necessary in the preceding theorem because K2 is a connected regular bipartite graph, but K2 ̸∈ CCR. Corollary 2.8. Membership in the class of CCR graphs is polynomially verifiable. Proof. Finding maximum or perfect matchings is a well-known polynomial procedure and hence so is checking 1-extendability and 1-expandability. But then the result follows immediately by Theorem 2.7. Corollary 2.9. If G = (A, B) is a bipartite graph with (a) |A| = |B| and G contains a Hamiltonian cycle or (b) |A| < |B| and G contains a Hamiltonian path, then G ∈ CCR. Proof. Part (a) is obvious since it is clear that every Hamiltonian bipartite graph has the property that every edge of G lies in a perfect matching; i.e., G is 1-extendable and then the result follows by Theorem 2.7. Suppose then that |A| < |B|. Then any Hamiltonian path must begin and end in B. Hence this path P is, in particular, a spanning forest for G such that degP (a) = 2, for all a ∈ A. Then G is 1-expanding by Corollary 6.2.2 of [1] and again the result follows by Theorem 2.7. Remark. The hypothesis that |A| < |B| is necessary in Corollary 2.9(b) since, for example, no path of odd length belongs to CCR. Remark. There exist connected regular bipartite graphs which are not Hamiltonian. For example, Horton (cf. p. 240 of [2]) discovered the first connected cubic bipartite non-Hamiltonian graph. It contains 96 vertices. The smallest such graphs known to date are due to Georges [3] and independently to Kelmans [11] and possess 50 vertices. On the other hand, if G is a 2-connected k-regular bipartite graph on at most 6k − 38 vertices, then it is Hamiltonian [8].
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Fig. 2.3. Two examples.
Remark. There are balanced bipartite graphs which contain perfect matchings, but which are not 1-extendable and hence do not belong to CCR. We present some examples. Let A1 = {a1 , . . . , an }, B1 = {b1 , . . . , bn }, A′1 = {a′1 , . . . , a′n } and B′1 = {b′1 , . . . , b′n }. Form an n-connected balanced bipartite graph Gn on 4n vertices having bipartition (A1 ∪ A′1 , B1 ∪ B′1 ) by taking the union of three complete bipartite graphs having bipartitions (A1 , B1 ), (B1 , A′1 ) and (A′1 , B′1 ). Then Gn clearly has a perfect matching, but fails to be 1-extendable, for any edge of the form bi a′j fails to lie in any perfect matching. Clearly, ′ α(Gn /b i aj ) = α(Gn ) = 2n, as well, so Gn is not contraction-critical.
Theorem 2.10. Let G = (A, B) be a connected bipartite graph with |A| ≤ |B|. Let ∆(G) denote the maximum degree among all vertices in G. Then if deg (a) = ∆(G) for all vertices a ∈ A, G ∈ CCR. Proof. By König’s edge coloring theorem (cf. Lemma 1.4.18; [15]), χ ′ (G) = ∆(G). So the χ ′ edge color classes represent a partition in which each vertex of A is incident with precisely one edge from each of the ∆ color classes. So every edge of G lies in a complete matching of A into B and hence G is 1-expanding with |A| < |B| or 1-extendable. Again, then, the result follows from Theorem 2.7. Definition. A bipartite graph G with bipartition (A, B) is biregular if deg(a) is constant over all vertices a ∈ A and deg(b) is constant over all vertices b ∈ B. The next result follows immediately from Theorem 2.10. Corollary 2.11. If G is a connected biregular or regular bipartite graph of order at least 3, then G ∈ CCR. Remark. Suppose G = (A, B) is a connected bipartite graph in which deg(a) has a constant value m over A, but suppose there is a vertex b ∈ B with deg(b) = ∆(G) = ∆ > m. Is G necessarily in CCR? The answer is ‘‘no’’ in both the balanced and unbalanced cases. In both graphs G1 and G2 shown in Fig. 2.3, each vertex in A has degree 3, and the same holds in G1 − a and G2 − a. Both G1 and G1 − a have ∆ = 4, whereas both G2 and G2 − a have ∆ = 5. Graphs G1 and G1 − a are in CCR, whereas G2 and G2 − a are not in CCR. (The edge e is not contractible in either G2 or G2 − a.) 3. Contraction-critical trees Definition. A tree T (T ̸= K1 ) is called an ED-tree if for every pair of endvertices a and b in T , d(a, b) is even. Let T be a tree of order at least two and let (A, B) be its bipartition. Then for u, v ∈ V (T ), d(u, v) is even if and only if {u, v} ⊆ A or {u, v} ⊆ B. Therefore, T is an ED-tree if and only if one of A and B contains all the endvertices of T . Theorem 3.1 is part of Theorem 12 of [4]. It was also proved later in [7] in which the authors call an ED-tree a strong independence tree. Theorem 3.1. If T is an ED-tree with bipartition (A, B), where B contains all the endvertices of T , then B is the unique maximum independent set in T . We can now characterize those trees belonging to CCR. Theorem 3.2. Let T be a tree. Then T ∈ CCR if and only if T is an ED-tree of order at least 3. Proof. Suppose T ∈ CCR. Let (A, B) be the bipartition of T , where we may assume that |A| ≤ |B|. Note that |V (T )| ≥ 3 since K2 is not in CCR. Assume first that |A| = |B|. Then by Theorem 2.7(c), T is 1-extendable. Let v be a vertex of degree 1 in T and let X = {v}. By symmetry, we may assume that v ∈ A. Since |V (T )| ≥ 3 and |A| = |B|, |A| ≥ 2 and hence X is a proper nonempty subset of A. Moreover, since v is a vertex of degree 1, |N (X )| = |X | = 1. But this contradicts Lemma 1.2 and hence we
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Fig. 4.1. A lattice of implications for balanced bipartite graphs (dashed arrows signify non-implications).
have |A| < |B|. Then by Theorem 2.7(c), B is a unique maximum independent set. So by the Remark following Lemma 2.4, B contains all the degree 1 vertices and hence T is an ED-tree. Now, conversely, suppose that T is an ED-tree of order at least 3. Let (A, B) be the bipartition of T where B contains all the vertices of degree 1 in T . By Theorem 3.1, B is the unique maximum independent set in T , so in particular, α(T ) = |B|. But then by Lemma 2.4(a) and the fact that T ̸= K2 , T ∈ CCR. 4. Closing remarks We conclude with some remarks relating the class CCR to well-covered graphs. Since by Lemma 2.1, CCR graphs must be bipartite, we will restrict ourselves to bipartite graphs. Moreover, since well-covered bipartite graphs must be balanced (cf. Theorem 4.1 [18]), we will assume our graphs to be balanced. We assume that all graphs under discussion are connected. Bipartite well-covered graphs have been characterized by Ravindra. To state his characterization, we introduce the following notation. Let G = (A, B) be a balanced bipartite graph containing a perfect matching F and let e = uv ∈ E (G) be an edge in F . Then Ge denotes the induced subgraph on N (u) ∪ N (v). Theorem 4.1 ([18]). The following three statements are equivalent for a balanced bipartite graph G without isolated vertices: (a) G is well-covered. (b) G contains a perfect matching F such that for all e ∈ F , Ge is complete bipartite. (c) For every perfect matching F in G and every edge e ∈ F , Ge is complete bipartite. We now summarize some relationships to other graph properties in the following implication diagram. We will call property (b) in the above theorem the R-property (see Fig. 4.1). Counterexamples for the non-implications labeled A through E are as follows: A: the path P4 ; B: the cycle C6 ; C: the cube with the edges of a 4-cycle deleted; D and E: the path P6 . The fact that the presence of a perfect matching in the balanced bipartite graph G implies that α(G) = |V (G)|/2 follows from the König minimax theorem for bipartite graphs and a wellknown Gallai identity. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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Note
A note on graphs contraction-critical with respect to independence number Michael D. Plummer a,∗ , Akira Saito b a
Department of Mathematics, Vanderbilt University, Nashville, TN 37240, United States
b
Department of Information Science, Nihon University, Sakurajosui 3-25-40, Setagaya-ku, Tokyo 156-8550, Japan
article
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Article history: Received 1 August 2013 Received in revised form 4 February 2014 Accepted 5 February 2014 Available online 13 March 2014 Keywords: Independent set Contraction-critical Matching 1-extendable Polynomial algorithm
abstract Let α(G) denote the independence number of graph G, i.e., the size of any maximum independent set of vertices. A graph G is contraction-critical (with respect to α ) if α(G/ xy) < α(G), for every edge xy ∈ E (G), where G/ xy denotes the graph obtained from G by shrinking the edge xy to a single vertex xy and deleting the loop thus formed. Let us denote the class of all such contraction-critical graphs by CCR. First it is shown that all graphs in CCR must be bipartite. Let G = (A, B) be a bipartite graph with bipartition A ∪ B. It is then shown that (a) if |A| < |B|, then G ∈ CCR if and only if B is the unique maximum independent set in G if and only if G is 1-expanding, while (b) if |A| = |B|, then G ∈ CCR if and only if A and B are the only two maximum independent sets in G if and only if G is 1-extendable. It follows that CCR graphs can be recognized in polynomial time. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Let α(G) denote the independence number of the graph G, that is, the size of any largest independent set of vertices in G. Problems involving α(G) have proven to be quite difficult on the whole so far. Perhaps this is not so surprising in view of the fact that the determination of α(G) for arbitrary G was one of the first problems to be shown to be NP-complete [10]. On the other hand, for certain special subclasses of graphs, α(G) can be computed in polynomial time. For example, if G is bipartite, then α(G) may be computed in polynomial time by using a polynomial matching algorithm together with two classical results by König and Gallai (cf. [15]). Claw-free graphs constitute a second graph class admitting a polynomial algorithm for determining α (cf. [16,20]). Clearly, if a graph has the property that every maximal independent set is, in fact, maximum (the so-called well-covered graphs), determination of α(G) becomes trivially polynomial, but no polynomial recognition algorithm for well-covered graphs is known. Interesting results about well-covered graphs have been obtained, however. For a sampling, see the surveys [5,17]. Historically, useful information about certain graph parameters has been gleaned from the study of various kinds of ‘‘criticality’’ with respect to the given parameter. For example, graphs which are critical with respect to edge deletion (the so-called α -critical graphs) have been investigated. These are the graphs G for which α(G − e) > α(G), for every edge e ∈ E (G). However, no polynomial recognition algorithm for this graph family is known and these graphs remain far from well-understood. For more on this graph family, see Section 12.1 of [15].
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Corresponding author. E-mail addresses: [email protected], [email protected] (M.D. Plummer), [email protected] (A. Saito).
http://dx.doi.org/10.1016/j.disc.2014.02.004 0012-365X/© 2014 Elsevier B.V. All rights reserved.
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The idea of criticality with respect to edge-contraction has proved extremely useful in the study of certain graph properties, notably in the study of graph connectivity (cf. [12]), graph coloring (cf. [9,13,21]) and graph minors (cf. [19]). However, the notion of criticality of edge contraction applied to independence number does not seem to have been investigated heretofore. Let G be a graph and xy an edge in G. The edge xy is be said to be contraction-critical (with respect to α ) if α(G/ xy) < α(G), where G/ xy denotes the graph obtained from G by shrinking the edge xy to a single vertex xy and deleting the loop and any multiple edges thus formed. A graph G is contraction-critical if every edge of G is contraction-critical. Let us denote the class of all such graphs by CCR. (Note that a graph belongs to CCR if and only if all its components belong to CCR.) In the present note, we show, that the CCR graphs turn out to be a well-known polynomially recognizable family in disguise. First we show that all members of the family CCR must be bipartite and then we obtain a characterization theorem. To accomplish this, we shall need the notions of 1-extendability and 1-expandability both of which deal with matchings. Although these two properties may be defined for graphs in general, we will confine both properties to bipartite graphs since the class CCR is bipartite. So let G = (A, B) be a bipartite graph with bipartition A ∪ B where, without loss of generality, we assume that |A| ≤ |B|. Definition ([1]). If G = (A, B) is bipartite with |A| < |B|, then G is 1-expanding if for every edge e of G, there is a matching of all A into B which contains e. An equivalent version of this property is given by the following result. Lemma 1.1 (Lemma 6.2.4 [1]). A connected bipartite graph G = (A, B) with |A| < |B| is 1-expanding if and only if for all X ⊆ A, ∅ ̸= X ̸= A, |N (X )| ≥ |X | + 1. The ‘‘balanced’’ version of the above property is the following. Definition ([6,15]). If G = (A, B) is bipartite with |A| = |B|, then G is 1-extendable if every edge of G lies in a perfect matching in G. This property too admits a useful equivalent statement. Lemma 1.2 ([6,14]). A connected bipartite graph G = (A, B) with |A| = |B| is 1-extendable if and only if for all X ⊆ A, ∅ ̸= X ̸= A, |N (X )| ≥ |X | + 1. In this note we first show that if G ∈ CCR, it must be bipartite. We then obtain the following characterizations: (a) if G = (A, B) is a connected bipartite graph with |A| < |B|, then G ∈ CCR if and only if B is the unique maximum independent set in G if and only if G is 1-expanding, while (b) if G = (A, B) is a connected bipartite graph with |A| = |B|, then G ∈ CCR if and only if A and B are the only two maximum independent sets in G if and only if G is 1-extendable. It follows from these characterizations that, since both the 1-extendable and 1-expanding properties can be checked in polynomial time via matching algorithms, membership in the class CCR is also polynomially verifiable. We shall denote by ν(G) the size of a maximum matching in graph G and by τ (G) the size of a minimum vertex cover in G. Note that we will not consider the single-vertex graph K1 to be bipartite. For all other terminology and notation, see [2]. Note also that in this paper all graphs are finite and simple. 2. Main results Lemma 2.1. If G belongs to CCR, then G is a bipartite graph and every maximum independent set is a partite set of G. In particular, |V (G)| ≤ 2α(G). Proof. Let G be a contraction-critical graph and let I be a maximum independent set in G. If G − I contains an edge e, let G′ = G/ˆe. Then I is an independent set in G′ which implies that α(G′ ) ≥ α(G). This contradicts the definition of a contractioncritical graph. Therefore, V (G) − I is an independent set. Since I is also independent, G is a bipartite graph with bipartition (I , V (G) − I ). Since I is a maximum independent set and V (G) − I is an independent set, |V (G) − I | ≤ |I | = α(G) and hence |V (G)| ≤ 2α(G). The following result is well-known. Lemma 2.2. If G is a connected bipartite graph with bipartition (A, B), then this bipartition is unique.
Corollary 2.3. Suppose G ∈ CCR with bipartition (A, B), where by Lemma 2.1 we may assume |B| = α(G). Then if |A| < |B|, G contains exactly one maximum independent set, namely B, and if |A| = |B|, G contains exactly two maximum independent sets, namely A and B.
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Fig. 2.1. The edge e is not contraction-critical.
Fig. 2.2. The edge e is not contraction-critical.
Proof. Let I be a maximum independent set in G. Then (I , V (G) − I ) is a bipartition of G by Lemma 2.1 and by Lemma 2.2, this bipartition is unique. The result follows. Remark. A bipartite graph can be connected and have exactly two maximum independent sets, but fail to be contractioncritical. An example is given in Fig. 2.1 where α = 5 and the edge e is not contraction-critical. Lemma 2.4. Suppose that G is a connected bipartite graph with bipartition (A, B) and suppose G ̸= K2 . (a) If |A| < |B| and B is the only maximum independent set in G, then G ∈ CCR. (b) If |A| = |B| and A and B are the only maximum independent sets in G, then G ∈ CCR. Proof. (a) Suppose G is a connected bipartite graph with |A| < |B| and that B is the only maximum independent set in G, ) = α(G). but suppose G ̸∈ CCR. So there is an a ∈ A and a b ∈ B with a adjacent to b such that α(G/ab ˆ ˆ Let I be a maximum independent set in G/ab. Note that |I | = α(G/ab) = α(G). If ab ̸∈ ˆI, then ˆI is an independent set in G and hence a maximum independent set in G. But this is a contradiction since b ∈ B − ˆI and hence B ̸= ˆI. Thus we have ∈ ˆI. Let Ia = (ˆI − {ab }) ∪ {a}. Then Ia is a maximum independent set in G, since |I | = |ˆI | = α(G). But this is again a ab contradiction since a ∈ Ia − B and hence Ia ̸= B. (b) If |A| = 1, then since |B| = |A| and G is connected, we have that G = K2 . This contradicts the assumption. Hence we may suppose both |A| ≥ 2 and |B| ≥ 2. Again, as in Case (a), suppose G ̸∈ CCR. So there exist vertices a ∈ A and b ∈ B such that a and b are adjacent, but ) = α(G). α(G/ab
lies in every maximum independent set in G/ab. Claim 1. Vertex ab but ab ̸∈ ˆI. Now ˆI ⊆ V (G) and |ˆI | = α(G/ab ) = α(G) Suppose, to the contrary, that ˆI is a maximum independent set in G/ab, and so ˆI is a maximum independent set in G. But then either ˆI = A or ˆI = B. Without loss of generality, suppose ˆI = B. But b ∈ B and b ̸∈ ˆI, which is a contradiction. This proves Claim 1. then ˆI ∩ A ̸= ∅ and ˆI ∩ B ̸= ∅. Claim 2. If ˆI is a maximum independent set in G/ab, }) ∪ {b} = I ′ ⊆ B. By symmetry, it is enough to show that ˆI ∩ A ̸= ∅. Suppose, to the contrary, that ˆI ∩ A = ∅. Hence (ˆI − {ab But |I ′ | = α(G), so I ′ = B. (I ′ ̸= A, since b ∈ I ′ .) But |B| ≥ 2, so I ′ − {b} ̸= ∅. So I ′′ = (I ′ − {b}) ∪ {a} is independent in G and hence |I ′′ | = α(G). So |I ′′ | = |I ′ | = α(G) and I ′′ ∩ B ̸= ∅ ̸= I ′′ ∩ A. But then I ′′ is a maximum independent set in G, but A ̸= I ′′ ̸= B, which is a contradiction. This completes the proof of Claim 2. and hence ab ∈ ˆI. Let I0 = (ˆI −{ab })∪{a}. Then I0 is independent Again, suppose ˆI is a maximum independent set in G/ab in G and |I0 | = |ˆI | = α(G) and so I0 is a maximum independent set in G. But I0 ∩ A ̸= ∅ ̸= I0 ∩ B by Claim 2. So I0 ̸= A and I0 ̸= B, which is a contradiction. Let us point out, however, that if G = (A, B) is a connected bipartite graph where |A| ≤ |B| and if G has a unique maximum independent set I, it is not necessarily true that I = B nor is it necessarily true that G ∈ CCR. (See the bipartite graph in Fig. 2.2 in which I = {a2 , a3 , b2 , b3 , b4 } is the unique maximum independent set, I ̸= B and edge e = a1 b1 is not contraction-critical.)
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However, the next remark is clear. Remark. Every graph has a maximum independent set that contains all of its degree 1 vertices. In particular, if a graph G has a unique maximum independent set, it contains all its degree 1 vertices. Theorem 2.5. Suppose G = (A, B) is a connected bipartite graph. (a) If |A| = |B| and G is 1-extendable, then A and B are the unique maximum independent sets in G. (b) If |A| < |B| and G is 1-expanding, then B is the unique maximum independent set in G. Proof. Let I be a maximum independent set in G and let IA = I ∩ A and IB = I ∩ B. If IA ̸= ∅, and IA ̸= A, let J = IB ∪ N (IA ). Then J ⊆ B and hence J is independent. Moreover, since I is independent, IB ∩ N (IA ) = ∅, and since G is a 1-extendable graph or a 1-expanding graph with |A| < |B|, by Lemmas 1.1 and 1.2 it follows that |N (IA )| ≥ |IA | + 1. However, this implies |J | = |IB | + |N (IA )| ≥ |IA | + |IB | + 1 = |I | + 1, which contradicts the maximality of I. Therefore, we have IA = ∅ or IA = A. If IA = A, then IB = ∅ since G is connected, and hence I = A. If IA = ∅, then I ⊆ B, and by the maximality of I, we have I = B. Therefore, we have either I = A or I = B, which proves (a). If |A| < |B|, then A cannot be a maximum independent set, and I = B, which proves (b). The next corollary is immediate by Theorem 2.5 and Lemma 2.4. Corollary 2.6. If G = (A, B) is a 1-expanding or 1-extendable connected bipartite graph of order at least 3, then G ∈ CCR.
Now we assemble the above results into a characterization theorem for graphs in CCR. Theorem 2.7. Let G be a connected graph, G ̸= K2 . Then the following are equivalent: (a) G ∈ CCR. (b) G is bipartite with bipartition (A, B) where we may assume |A| ≤ |B| and if |A| < |B|, B is the only maximum independent set in G, whereas if |A| = |B|, A and B are the only maximum independent sets in G. (c) G is bipartite with bipartition (A, B) where we may assume |A| ≤ |B| and if |A| < |B|, G is 1-expanding whereas if |A| = |B|, G is 1-extendable. Proof. Suppose G ∈ CCR. Then by Lemma 2.1, G is bipartite and if (A, B) is the bipartition of G, we may assume B is a maximum independent set in G. Then by Corollary 2.3, part (b) follows. Suppose next that part (b) holds. We prove part (c) in the case when |A| < |B|; the proof when |A| = |B| is quite similar. Suppose, to the contrary, that G is not 1-expanding. Then there is a subset X of A, ∅ ̸= X ̸= A, such that |N (X )| ≤ |X | by Lemma 1.1. But then the set S = (B − N (X )) ∪ X is independent and |S | ≥ |B|. Hence |S | = |B| and S is a maximum independent set. But S ̸= B, which is a contradiction. So part (c) holds. Suppose now that part (c) holds. Then part (b) holds by Theorem 2.5. Finally if part (b) holds, then part (a) holds by Lemma 2.4. Remark. The assumption that G ̸= K2 is necessary in the preceding theorem because K2 is a connected regular bipartite graph, but K2 ̸∈ CCR. Corollary 2.8. Membership in the class of CCR graphs is polynomially verifiable. Proof. Finding maximum or perfect matchings is a well-known polynomial procedure and hence so is checking 1-extendability and 1-expandability. But then the result follows immediately by Theorem 2.7. Corollary 2.9. If G = (A, B) is a bipartite graph with (a) |A| = |B| and G contains a Hamiltonian cycle or (b) |A| < |B| and G contains a Hamiltonian path, then G ∈ CCR. Proof. Part (a) is obvious since it is clear that every Hamiltonian bipartite graph has the property that every edge of G lies in a perfect matching; i.e., G is 1-extendable and then the result follows by Theorem 2.7. Suppose then that |A| < |B|. Then any Hamiltonian path must begin and end in B. Hence this path P is, in particular, a spanning forest for G such that degP (a) = 2, for all a ∈ A. Then G is 1-expanding by Corollary 6.2.2 of [1] and again the result follows by Theorem 2.7. Remark. The hypothesis that |A| < |B| is necessary in Corollary 2.9(b) since, for example, no path of odd length belongs to CCR. Remark. There exist connected regular bipartite graphs which are not Hamiltonian. For example, Horton (cf. p. 240 of [2]) discovered the first connected cubic bipartite non-Hamiltonian graph. It contains 96 vertices. The smallest such graphs known to date are due to Georges [3] and independently to Kelmans [11] and possess 50 vertices. On the other hand, if G is a 2-connected k-regular bipartite graph on at most 6k − 38 vertices, then it is Hamiltonian [8].
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Fig. 2.3. Two examples.
Remark. There are balanced bipartite graphs which contain perfect matchings, but which are not 1-extendable and hence do not belong to CCR. We present some examples. Let A1 = {a1 , . . . , an }, B1 = {b1 , . . . , bn }, A′1 = {a′1 , . . . , a′n } and B′1 = {b′1 , . . . , b′n }. Form an n-connected balanced bipartite graph Gn on 4n vertices having bipartition (A1 ∪ A′1 , B1 ∪ B′1 ) by taking the union of three complete bipartite graphs having bipartitions (A1 , B1 ), (B1 , A′1 ) and (A′1 , B′1 ). Then Gn clearly has a perfect matching, but fails to be 1-extendable, for any edge of the form bi a′j fails to lie in any perfect matching. Clearly, ′ α(Gn /b i aj ) = α(Gn ) = 2n, as well, so Gn is not contraction-critical.
Theorem 2.10. Let G = (A, B) be a connected bipartite graph with |A| ≤ |B|. Let ∆(G) denote the maximum degree among all vertices in G. Then if deg (a) = ∆(G) for all vertices a ∈ A, G ∈ CCR. Proof. By König’s edge coloring theorem (cf. Lemma 1.4.18; [15]), χ ′ (G) = ∆(G). So the χ ′ edge color classes represent a partition in which each vertex of A is incident with precisely one edge from each of the ∆ color classes. So every edge of G lies in a complete matching of A into B and hence G is 1-expanding with |A| < |B| or 1-extendable. Again, then, the result follows from Theorem 2.7. Definition. A bipartite graph G with bipartition (A, B) is biregular if deg(a) is constant over all vertices a ∈ A and deg(b) is constant over all vertices b ∈ B. The next result follows immediately from Theorem 2.10. Corollary 2.11. If G is a connected biregular or regular bipartite graph of order at least 3, then G ∈ CCR. Remark. Suppose G = (A, B) is a connected bipartite graph in which deg(a) has a constant value m over A, but suppose there is a vertex b ∈ B with deg(b) = ∆(G) = ∆ > m. Is G necessarily in CCR? The answer is ‘‘no’’ in both the balanced and unbalanced cases. In both graphs G1 and G2 shown in Fig. 2.3, each vertex in A has degree 3, and the same holds in G1 − a and G2 − a. Both G1 and G1 − a have ∆ = 4, whereas both G2 and G2 − a have ∆ = 5. Graphs G1 and G1 − a are in CCR, whereas G2 and G2 − a are not in CCR. (The edge e is not contractible in either G2 or G2 − a.) 3. Contraction-critical trees Definition. A tree T (T ̸= K1 ) is called an ED-tree if for every pair of endvertices a and b in T , d(a, b) is even. Let T be a tree of order at least two and let (A, B) be its bipartition. Then for u, v ∈ V (T ), d(u, v) is even if and only if {u, v} ⊆ A or {u, v} ⊆ B. Therefore, T is an ED-tree if and only if one of A and B contains all the endvertices of T . Theorem 3.1 is part of Theorem 12 of [4]. It was also proved later in [7] in which the authors call an ED-tree a strong independence tree. Theorem 3.1. If T is an ED-tree with bipartition (A, B), where B contains all the endvertices of T , then B is the unique maximum independent set in T . We can now characterize those trees belonging to CCR. Theorem 3.2. Let T be a tree. Then T ∈ CCR if and only if T is an ED-tree of order at least 3. Proof. Suppose T ∈ CCR. Let (A, B) be the bipartition of T , where we may assume that |A| ≤ |B|. Note that |V (T )| ≥ 3 since K2 is not in CCR. Assume first that |A| = |B|. Then by Theorem 2.7(c), T is 1-extendable. Let v be a vertex of degree 1 in T and let X = {v}. By symmetry, we may assume that v ∈ A. Since |V (T )| ≥ 3 and |A| = |B|, |A| ≥ 2 and hence X is a proper nonempty subset of A. Moreover, since v is a vertex of degree 1, |N (X )| = |X | = 1. But this contradicts Lemma 1.2 and hence we
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Fig. 4.1. A lattice of implications for balanced bipartite graphs (dashed arrows signify non-implications).
have |A| < |B|. Then by Theorem 2.7(c), B is a unique maximum independent set. So by the Remark following Lemma 2.4, B contains all the degree 1 vertices and hence T is an ED-tree. Now, conversely, suppose that T is an ED-tree of order at least 3. Let (A, B) be the bipartition of T where B contains all the vertices of degree 1 in T . By Theorem 3.1, B is the unique maximum independent set in T , so in particular, α(T ) = |B|. But then by Lemma 2.4(a) and the fact that T ̸= K2 , T ∈ CCR. 4. Closing remarks We conclude with some remarks relating the class CCR to well-covered graphs. Since by Lemma 2.1, CCR graphs must be bipartite, we will restrict ourselves to bipartite graphs. Moreover, since well-covered bipartite graphs must be balanced (cf. Theorem 4.1 [18]), we will assume our graphs to be balanced. We assume that all graphs under discussion are connected. Bipartite well-covered graphs have been characterized by Ravindra. To state his characterization, we introduce the following notation. Let G = (A, B) be a balanced bipartite graph containing a perfect matching F and let e = uv ∈ E (G) be an edge in F . Then Ge denotes the induced subgraph on N (u) ∪ N (v). Theorem 4.1 ([18]). The following three statements are equivalent for a balanced bipartite graph G without isolated vertices: (a) G is well-covered. (b) G contains a perfect matching F such that for all e ∈ F , Ge is complete bipartite. (c) For every perfect matching F in G and every edge e ∈ F , Ge is complete bipartite. We now summarize some relationships to other graph properties in the following implication diagram. We will call property (b) in the above theorem the R-property (see Fig. 4.1). Counterexamples for the non-implications labeled A through E are as follows: A: the path P4 ; B: the cycle C6 ; C: the cube with the edges of a 4-cycle deleted; D and E: the path P6 . The fact that the presence of a perfect matching in the balanced bipartite graph G implies that α(G) = |V (G)|/2 follows from the König minimax theorem for bipartite graphs and a wellknown Gallai identity. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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