key: cord-0044036-zzrakyw2
authors: Hasunuma, Toru
title: Connectivity Keeping Trees in 2-Connected Graphs with Girth Conditions
date: 2020-04-30
journal: Combinatorial Algorithms
DOI: 10.1007/978-3-030-48966-3_24
sha: 6f79bb513b90d27528e1dcd949252e4faf3d1912
doc_id: 44036
cord_uid: zzrakyw2

Mader conjectured in 2010 that for any tree T of order m, every k-connected graph G with minimum degree at least [Formula: see text] contains a subtree [Formula: see text] such that [Formula: see text] is k-connected. This conjecture has been proved for [Formula: see text]; however, it remains open for general [Formula: see text]; for [Formula: see text], partially affirmative answers have been shown, all of which restrict the class of trees to special subclasses such as trees of order at most 8, trees with diameter at most 4, trees with at most 5 internal vertices, and caterpillars. Instead of restricting the class of trees, we consider 2-connected graphs with girth conditions. We then show that Mader’s conjecture is true for every 2-connected graph G with [Formula: see text], where g(G) and [Formula: see text] denote the girth of G and the minimum degree of a vertex in G, respectively. Besides, we show that for every 2-connected graph G with [Formula: see text], the lower bound of [Formula: see text] on [Formula: see text] in Mader’s conjecture can be improved to [Formula: see text] if [Formula: see text]. Moreover, the lower bound of [Formula: see text] (respectively, [Formula: see text]) on g(G) in these results can be improved to [Formula: see text] (respectively, [Formula: see text] with [Formula: see text]) if no six (respectively, four) cycles of length g(G) have a common path of length [Formula: see text] in G. Mader’s conjecture is interesting not only from a theoretical point of view but also from a practical point of view, since it may be applied to fault-tolerant problems in communication networks. Our proofs lead to [Formula: see text] time algorithms for finding a desired subtree in a given 2-connected graph G satisfying the assumptions.

Throughout this paper, a graph G = (V, E) means a simple undirected graph unless stated otherwise. The minimum degree of a vertex in G is denoted by δ(G). For a proper subset S V (G), we denote by G − S the graph obtained from G by deleting every vertex in S, where G − {v} is abbreviated to G − v. For two sets A and B, we denote by A \ B the set difference {x | x ∈ A, x ∈ B}.

For a nonempty subset S ⊆ V (G), the subgraph of G induced by S is denoted by S G , i.e., S G = G − (V (G) \ S).

In 1972, Chartrand, Kaigars, and Lick proved the following result on the existence of a vertex whose removal does not influence k-connectedness of a graph.

Theorem 1 (Chartrand, Kaigars, and Lick [1] ). Every k-connected graph G with δ(G) ≥ 3k 2 contains a vertex v such that G − v is k-connected.

After more than 30 years, Fujita and Kawarabayashi considered a similar problem for an edge of a graph and showed the following.

Theorem 2 (Fujita and Kawarabayashi [3] ). Every k-connected graph G with δ(G) ≥ 3k 2 + 2 contains an edge uv such that G − {u, v} is k-connected.

In the same paper, they conjectured the next statement.

There is a function f (m) such that every k-connected graph G with δ(G) ≥ 3k 2 + f (m) contains a connected subgraph W of order m such that G − V (W ) is k-connected.

Note that the condition that W is connected is essential, since by iteratively applying Theorem 1, we can see that every k-connected graph G with δ(G) ≥ 3k 2 +m−1 contains a subgraph X of order m such that G−V (X) is k-connected. In 2010, Mader [8] settled Conjecture 1 by showing the following result. Mader's result in fact improves the lower bound on δ(G) in Theorem 2 and generalizes Theorem 1.

Theorem 3 (Mader [8] ). Every k-connected graph G with δ(G) ≥ 3k 2 + m − 1 contains a path P of order m such that G − V (P ) is k-connected.

Based on this result, Mader conjectured the following, i.e., a path in Theorem 3 can be generalized to any tree of the same order.

Conjecture 2 (Mader [8] ). For any tree T of order m, every k-connected graph G with δ(G) ≥ 3k 2 + m − 1 contains a subtree T ∼ = T such that G − V (T ) is k-connected.

Mader's conjecture is a generalization not only from Theorem 1 but also from the next well-known result on the existence of a subtree isomorphic to any given tree. Apart from Mader's conjecture, Locke's conjecture concerning nonseparating trees in connected graphs is known. A k-cohesive graph is a non-trivial connected graph in which for any two distinct vertices u and v, the sum of the degrees of u and v and the distance between u and v is at least k.

Conjecture 3 (Locke [5] ). For any tree T of order m ≥ 3, every 2m-cohesive graph G has a subtree T ∼ = T such that G − V (T ) is connected.

Motivated by Locke's conjecture, Diwan and Tholiya proved a theorem which is weaker than the conjecture, but it is the same as Mader's conjecture for k = 1 (Mader in fact mentioned their result in the paper [8] ). Note that if G is connected and δ(G) ≥ m, then G is 2m-cohesive.

Theorem 4 (Diwan and Tholiya [2] ). For any tree T of order m, every con-

For general k ≥ 2, Mader's conjecture remains open; however for k = 2, partially affirmative answers have been shown. Tian et al. [10] first proved that Mader's conjecture for k = 2 is true when T is a star or a double-star, and they [11] further extended their results to a path-star or a path-double-star. Hasunuma and Ono [4] showed that for any tree T of order m,

is the number of internal vertices of T . As a corollary, it follows that Mader's conjecture for k = 2 holds for any tree T with n(T ) ≤ 5 and for any tree of order m ≤ 8. Lu and Zhang [6] also proved that Mader's conjecture for k = 2 is true for any tree with diameter at most 4. Very recently, it was reported that Lu and Ye [7] proved that Mader's conjecture for k = 2 holds for any caterpillars. Note that every known result which is a partially affirmative answer to Mader's conjecture for k = 2 restricts the class of trees to special subclasses. In this paper, we employ another approach to Mader's conjecture for k = 2. Namely, we add girth conditions to 2-connected graphs. The girth of a 2-connected graph G denoted by g(G) is the length of a smallest cycle in G. We then show that Mader's conjecture is true for every 2connected graph G with girth at least δ(G) − 6. Note that for any given integers r ≥ 2 and g ≥ 3, there exists an r-regular graph with girth g, which has been shown in [12] .

By increasing the lower bound of δ(G) − 6 on g(G), we can improve the lower bound of m + 2 on δ(G) to m + 1 if m ≥ 6. Namely, a stronger statement holds in such a case.

Moreover, by adding structural conditions, we can improve the girth conditions in Theorems 5 and 6.

For any tree T of order m, every 2-connected graph G with δ(G) ≥ m + 2 and g(G) ≥ δ(G) − 7 in which no six cycles of length g(G) have a common path of length g(G)

Theorem 8. For any tree T of order m ≥ 7, every 2-connected graph G with δ(G) ≥ m + 1 and g(G) ≥ δ(G) − 4 in which no four cycles of length g(G) have a common path of length g(G)

Mader's conjecture is interesting not only from a theoretical point of view but also from a practical point of view, since it may be applied to fault-tolerant problems in communication networks. That is, it is considered that Mader's conjecture guarantees the reliability of a communication network for a faulty subtree structure rather than a set of faulty vertices. Our proofs are constructive and lead to O(|V (G)| 4 ) time algorithms for finding a desired subtree in a given 2-connected graph G in Theorems 5 and 6 (respectively, Theorems 7 and 8) if

This paper is organized as follows. Section 2 presents notations, terminology, and known results used in this paper. Section 3 gives an outline of our proofs. Detailed proofs of Theorems 5 and 6 (respectively, Theorems 7 and 8) are given in Sect. 4 (respectively, Sect. 5). Section 6 concludes the paper with several remarks.

For a nonempty subset E ⊆ E(G), we denote by G − E and E the graph obtained from G by deleting every edge in E and the edge-induced subgraph of G by E , respectively. For v ∈ V (G), we denote by N G (v) the set of neighbors of v in G, i.e., vertices adjacent to v in G. The cardinality of N G (v) may be written by deg

A component of G is a maximal connected subgraph of G, while a block of G is a maximal connected subgraph of G without a cut vertex. A cyclic block is a block with order at least 3. For a tree T , the set of internal vertices, i.e., vertices with degree at least two, is denoted by V I (T ), while the set of leaves, i.e., vertices with degree one, is denoted by

We denote by d G (u, v) the distance between two vertices u and v in a connected graph G. The eccentricity ecc Proposition 1 can be stated in a more general form as follows.

Lemma 1 [4] . Let T be a tree of order m and S a subtree obtained from T by deleting leaves adjacent to a vertex in

Since any tree T of order m with diam(T ) ≥ m − 2 is a caterpillar and Mader's conjecture holds for any caterpillars [7] , the following result is obtained.

An orientation D of a graph G is a directed graph obtained from G by replacing each edge by an arc (directed edge) with the same end-vertices. The

. If G has a vertex with odd degree, we can find a directed walk W connecting two vertices with odd degree, and by inductively applying a similar discussion for G − E(W ), we can see the following lemma holds. We here remark that Lemma 3 holds for multigraphs.

D such that |deg + D (v)−deg − D (v)| ≤ 1 for any v ∈ V (D).

In this section, we explain the outline of our constructive proofs and the time complexity for the algorithms based on the proofs.

Let T be a tree of order m. Let G be a 2-connected graph with δ(G) ≥ m + 2. From Proposition 1, G contains a subtree T ∼ = T . Let B be a maximum block in G − V (T ), i.e., a block with the maximum order among all the blocks in

be a shortest path among all the paths of G connecting two vertices in B such that every internal vertex is not in

. By the definition of P , w can be adjacent to at most three

Thus, we can find a block with order at least |V (B)| + 2.

i.e., we can find a block with order at least |V (B)| + 1. If we have a manipulation to find such a subtree T , then by applying the manipulations for t ≥ 4 or t = 3 iteratively, we finally obtain a desired subtree T , i.e., T ∼ = T such that G − V (T ) is 2-connected. Therefore, if we can show the following statement, then it is concluded that Mader's conjecture for k = 2 is true.

The above manipulations can be algorithmically described as follows. 

Next, we consider the case that a 2-connected graph G has no triangle, i.e., g(G) ≥ 4. In such a case, we can show a similar lemma using the following statement. Note that the minimum degree condition δ(G) ≥ m + 2 is replaced with δ(G) ≥ m + 1 ≥ 3.

Let T be a tree of order m ≥ 2 and G a 2-connected graph with δ(G) ≥ m+1 and g(G) ≥ 4. For any subtree T ∼ = T in G and a maximum block

Proof. We show that the algorithm for Lemma 4 also works well under the assumption that Statement 2 holds.

Let T ⊂ G such that T ∼ = T . Let B be a maximum block in G − V (T ). Since δ(G − V (T )) ≥ 1, it may happen that B is not a cyclic block, i.e., B is a block with two vertices. Note that if B is not a cyclic block, then B is not 2-connected. Suppose that B is not a cyclic block. Assume that B = G − V (T ). Then, |V (G)| = m + 2. Since δ(G) ≥ m + 1, G must be a complete graph with at least four vertices, which contradicts the girth condition that g(G) ≥ 4. Therefore, if B is not a cyclic block, then B = G − V (T ). Hence, in the case that G − V (T ) has no cyclic block, the algorithm does not incorrectly output a subtree in Step 3.

Let P = (v 1 , v 2 , . . . , v t ) be a shortest path between two vertices in B such that every internal vertex is not in B. Suppose that t ≥ 4. By the definition of P and the girth condition 

In order to show our main results, we prove the following lemma. Since δ(G) ≥ m + 1, it holds that for any vertex w ∈ V (H), 

can be employed as a desired subtree W . If h(S ) ≥ 3, by inductively applying similar manipulations to descendants of x, we can finally obtain a desired subtree W . Note that in each extension step, disjointness of the sets of new children for descendants of x is guaranteed by the girth condition g(G) ≥ diam(S ) + 1.

The assumption that v is neither a leaf of T nor a leaf of S can be realized by preferentially selecting a vertex in

, we apply the manipulations for constructing W in a depth-first search order for S . The selection process for new children of a descendant of x and the extension process from W to T can be done greedily. If V B (T ) \ (V L (T ) ∪ V L (S )) = ∅, then we can directly obtain either W or T . Therefore, a desired subtree T can finally be found in O(|E(G)|) time.

Lemma 6 is stronger than Statement 1 under the assumption that g(G) ≥ diam(T ) − 1. Therefore, by Lemmas 4 and 6, we have the following. T of order m, every 2 

For any 2-connected graph G, it holds that g(G) ≥ 3. Thus, the following result by Lu and Zhang [6] is obtained from Theorem 9.

Corollary 1 [6] . For any tree T of order m with diam(T ) ≤ 4,

Besides, by combining Lemmas 5 and 6, we have the following.

From Theorem 10, the following result for 2-connected graphs without a triangle is obtained. 4 (respectively, g(G) ≥ δ(G) − 3) . Note that such a restriction on g(G) for Theorem 5 follows from the fact that we use Lemma 2.

In this section, we try to improve the lower bounds on g(G) in Theorems 5 and 6, and show that such improvements are possible if a 2-connected graph G satisfies a structural property on the smallest cycles. Note that for any two cycles C 1 and T be a tree of order m. Let G be a 2-connected graph with δ(G) ≥ m + 2 and g(G) ≥ diam(T ) − 2 in which no six cycles of length g(G) have a common path of length g(G)

Proof. We use the notations such as T , B, v, H, S , C(u), W , and x with the same meaning in the proof of Lemma 6. If diam(S ) ≤ 2, then we can easily construct a desired subtree W without an additional structural property. Suppose that diam(S ) ≥ 3. By the discussion in the proof of Lemma 6, we suppose that v is not a leaf of S and there is no vertex w in H such that C(v) ⊆ N G (w). For u ∈ C(v), we denote by S u the subtree rooted at u in S . Let F be a component of H containing x. Note that |N G (F ) ∩ V (B)| ≤ 1. In the following discussion, w.l.o.g., we may assume that

Suppose that v is a pseudo-leaf of S and v is the non-leaf vertex adjacent to v in S . If there exists a vertex y in H such that v ∈ N G (y), then by letting the vertex y correspond to v, we can obtain a desired subtree W . If there is no vertex in H which is adjacent to v , then v ∈ V B (T ). Thus, we may assume that if diam(S ) ≥ 4, v is not a pseudo-leaf of S , and if diam(S ) = 3, the central vertices v, v are in V B (T ) such that {v, v } ∩ N G (w) = ∅ for any w ∈ V (H). Suppose that diam(S ) = 3. Let xy ∈ E(F ). Then,

The assumption on the smallest cycles implies that |N F (x) ∩ N F (y)| ≤ 5. Therefore, we can find y ∈ C(x) ⊂ N F (x) and

Thus, a desired subtree W can be constructed. In what follows, we suppose that diam(S ) ≥ 4.

It is sufficient to consider the case that g(G) = diam(T ) − 2 = diam(S ). Let P (S ) and Q(S ) be the set of peripheral vertices in S and the set of parents of a peripheral vertex in S , respectively. Let S = S −P (S ). Note that diam(S ) = diam(S ) − 2, and v ∈ P (S ) ∪ Q(S ) since any vertex in P (S ) is a leaf of S and any vertex in Q(S ) is a pseudo-leaf of S . For the subtree S , we apply the manipulations in the proof of Lemma 6. Let W be the subtree obtained after such manipulations and let W F = V (W ) ∩ V (F ) W . Suppose that {z 1 , z 2 , . . . , z q } is the set of vertices in W F which are corresponding to vertices in Q(S ). Note that q may be less than |Q(S )|. Let {u 1 , u 2 , . . . , u q } ⊆ Q(S ) such that u i is corresponding to

Suppose that |D(z k )| = r k + 1 for some k ∈ {1, 2, . . . , q}. Then |C(v) \ N G (z k )| = 1 and z k is adjacent to every vertex in T except for ones in

Instead of x, we let z k correspond to v and apply the manipulations in the proof of Lemma 6. Let W F be the resultant subtree in F . If w k is a pseudo-leaf of S , then we can immediately obtain a desired subtree W in this setting. Otherwise, there is no pseudo-leaf adjacent to v in S which corresponds to a vertex in the subtree W F . Thus, w.l.o.g., we may assume that u k ∈ C(v). Consider the case that u k ∈ V B (T ). Since u k is a pseudo-leaf of S , by the previous discussion, we may assume that

. This means that w k = p(p(u k )). Next consider the case that u k ∈ V B (T ). In this case, we may assume that no descendant of u k in T is in V B (T ). Hence, it is con-

can be employed as a desired subtree W . Consequently, we may assume that any vertex z i in {z 1 , z 2 , . . . , z q } satisfies that |D(z i )| ≥ r i + 2.

Let

Define I G as the (multi)graph with vertex set {z 1 , z 2 , . . . , z q } in which z i and z j are joined by |D (z i ) ∩ D (z j )| edges. Note that I G may be a multigraph only if diam(S ) = 4.

The assumption that no six cycles of length g(G) have a common path of length

in G implies that Δ(I G ) ≤ 5, i.e., each vertex in I G is incident to at most five edges. Besides, the intersection of at least seven (respectively, three) sets in {D (z 1 ), D (z 2 ), . . . , D (z q )} is empty if diam(S ) is even (respectively, odd). Modify the graph I G as follows, and let J G be the resultant (multi)graph.

1. Delete every edge generated by a vertex in the intersection of at least three sets D (z i1 ), D (z i2 ), and D (z i3 ). 2. Delete the edge generated by v B if v B is contained in exactly two sets D (z i1 ) and D (z i2 ).

Note that if v B is contained in exactly one set D (z i ), then |D(z i )| ≥ r i + 2 and |D(z j )| ≥ r j + 3 for any j = i. By Lemma 3, J G has an orientation D G such that

Note that if an orientation of J G satisfying the first condition does not satisfy the second condition, the reverse orientation satisfies both the conditions since Δ(I G ) ≤ 5. Based on D G , we can disjointly select r i vertices in D(z i ) for 1 ≤ i ≤ q as follows. For each arc from z i1 to z i2 in D G , we select the vertex in D(z i1 ) ∩ D(z i2 ) corresponding to the edge z i1 z i2 as a child of z i1 . Note that we do not select the vertex v B and any vertex in the intersection of at least three sets D (z i1 ), D (z i2 ), and D (z i3 ). In this way, we can appropriately extend W F for a desired subtree W and finally obtain a

Next, we consider the case that δ(G) ≥ m + 1. In this case, we need to strengthen the structural condition on the smallest cycles in Lemma 7. 

Proof. We use the notations in the proof of Lemma 7 with the same meaning. A desired subtree W can be constructed without an additional structural property if diam(S ) ≤ 2. Suppose that diam(S ) = 3. Applying a similar discussion in the proof of Lemma 7, we have that |N F (x)| ≥ |C(v)\N G (x)|+2 and |N F (y)\{x}| ≥ |C(v ) \ N G (y)| + 2. Since the condition on smallest cycles implies that |N F (x) ∩ N F (y)| ≤ 3, we can find y ∈ C(x) ⊂ N F (x) and C(y) ⊂ N F (y) \ {x} so that C(x) ∩ C(y) = ∅, |C(x)| = |C(v) \ N G (x)| and |C(y)| = |C(v ) \ N G (y)|. Suppose that diam(S ) ≥ 4. From a similar discussion in the proof of Lemma 7, we may assume that every vertex z i in {z 1 , z 2 , . . . , z q } satisfies that |D(z i )| ≥ r i + 1 and |D (z i )| ≥ r i + 2. Note that the degree condition δ(H ) ≥ u∈C(v)\{w} |V (S u )| is sufficient to construct a subtree U H in H . The assumption that no four cycles of length g(G) have a common path of length g(G)

Based on D G , we can disjointly select r i vertices in D(z i ) for 1 ≤ i ≤ q. Hence, we can appropriately extend W F in order to obtain a desired subtree T . From Lemmas 4, 5, 7, and 8, we have the following results. 

In this paper, we have shown that Mader's conjecture for k = 2 (with a weak degree condition δ(G) ≥ m + 1) holds for graphs with large girth. Mader's conjecture was posed in a purely mathematical interest; however, it has a potential application to fault-tolerant problems in communication networks. We then have shown that our constructive proofs lead to O(|V (G)| 4 ) time algorithms.

Our lower bounds on the girth in Theorems 5 and 7 can be improved if the upper bound on the diameter of a tree for which Mader's conjecture for k = 2 holds is improved. Namely, the following result follows from Theorem 9. In particular, by checking the proof in [7] , we can see that Statement 2 holds for any caterpillars; thus, the lower bounds on g(G) in Theorems 6 and 8 can be improved to δ(G) − 5 and δ(G) − 6, respectively. Besides, the restriction that g(G) ≥ δ(G)−4 (respectively, g(G) ≥ δ(G)−5) for an O(|V (G)| 4 ) time algorithm can be removed for Theorem 5 (respectively, Theorem 7). On the other hand, in order to improve the lower bounds on the girth in Theorems 9, 10, 11, and 12 directly, we may need some other techniques.

Even though Mader's conjecture for k = 2 still remains open, from Lemma 5 and Corollary 2, we may conjecture the following. Although we consider Mader's conjecture only for k = 2, it would be interesting to approach Mader's conjecture for general k ≥ 2 by considering girth conditions.

Critically n-connected graphs

Non-separating trees in connected graphs

Connectivity keeping edges in graphs with large minimum degree

Connectivity keeping trees in 2-connected graphs

Connectivity keeping trees in 2-connected graphs

Connectivity keeping caterpillars in 2-connected graphs

Connectivity keeping paths in k-connected graphs

Connectivity keeping trees in k-connected graphs

Connectivity keeping stars or double stars in 2-connected graphs

Nonseparating trees in 2-connected graphs and oriented trees in strongly connected digraphs

Connectivity in Graphs. Univ. of

Acknowledgments. The author is grateful to the reviewers for their helpful comments. This work was supported by JSPS KAKENHI Grant Number JP19K11829.